Recent zbMATH articles in MSC 54https://zbmath.org/atom/cc/542023-12-07T16:00:11.105023ZWerkzeugBook review of: T.-D. Bradley et al., Topology. A categorical approachhttps://zbmath.org/1522.000332023-12-07T16:00:11.105023Z"Ceniceros, Jose"https://zbmath.org/authors/?q=ai:ceniceros.joseReview of [Zbl 1517.54001].Tychonoff HED-spaces and Zemanian extensions of \(\mathsf{S4.3}\)https://zbmath.org/1522.030492023-12-07T16:00:11.105023Z"Bezhanishvili, Guram"https://zbmath.org/authors/?q=ai:bezhanishvili.guram"Bezhanishvili, Nick"https://zbmath.org/authors/?q=ai:bezhanishvili.nick"Lucero-Bryan, Joel"https://zbmath.org/authors/?q=ai:lucero-bryan.joel-gregory"Van Mill, Jan"https://zbmath.org/authors/?q=ai:van-mill.janSummary: We introduce the concept of a Zemanian logic above \(\mathsf{S4.3}\) and prove that an extension of \(\mathsf{S4.3}\) is the logic of a Tychonoff HED-space iff it is Zemanian.Equivalence relations invariant under group actionshttps://zbmath.org/1522.031182023-12-07T16:00:11.105023Z"Rzepecki, Tomasz"https://zbmath.org/authors/?q=ai:rzepecki.tomaszSummary: We extend some recent results about bounded invariant equivalence relations and invariant subgroups of definable groups: we show that type-definability and smoothness are equivalent conditions in a wider class of relations than heretofore considered, which includes all the cases for which the equivalence was proved before.
As a by-product, we show some analogous results in purely topological context (without direct use of model theory).Orders on magmas and computability theoryhttps://zbmath.org/1522.031722023-12-07T16:00:11.105023Z"Ha, Trang"https://zbmath.org/authors/?q=ai:ha.trang"Harizanov, Valentina"https://zbmath.org/authors/?q=ai:harizanov.valentina-sSummary: We investigate algebraic and computability-theoretic properties of orderable magmas. A magma is an algebraic structure with a single binary operation. A right order on a magma is a linear ordering of its domain, which is right-invariant with respect to the magma operation. We use tools of computability theory to investigate Turing complexity of orders on computable orderable magmas. A magma is computable if it is finite, or if its domain can be identified with the set of natural numbers and the magma operation is computable. Interesting orderable magmas that are not even associative come from knot theory.Warsaw discs and semicomputabilityhttps://zbmath.org/1522.031782023-12-07T16:00:11.105023Z"Iljazović, Zvonko"https://zbmath.org/authors/?q=ai:iljazovic.zvonko"Pažek, Bojan"https://zbmath.org/authors/?q=ai:pazek.bojanSummary: We examine conditions under which a semicomputable set in a computable metric space is computable. Topology plays an important role in the description of such conditions. Motivated by the known result that a semicomputable cell is computable if its boundary sphere is computable, we investigate semicomputable Warsaw discs and their boundary Warsaw circles. We prove that a semicomputable Warsaw disc is computable if its boundary Warsaw circle is semicomputable.On PCF spaces which are not Fréchet-Urysohnhttps://zbmath.org/1522.031842023-12-07T16:00:11.105023Z"Martínez, Juan Carlos"https://zbmath.org/authors/?q=ai:martinez.juan-carlosSummary: By means of a forcing argument, it was shown by \textit{L. Pereira} [Arch. Math. Logic 47, No. 5, 517--527 (2008; Zbl 1153.03025)] that if CH holds then there is a separable PCF space of height \(\omega_1 + 1\) which is not Fréchet-Urysohn. In this paper, we give a direct proof of Pereira's theorem [loc. cit.] by means of a forcing-free argument, and we extend his result to PCF spaces of any height \(\delta + 1\) where \(\delta < \omega_2\) with \(\operatorname{cf}(\delta) = \omega_1\).Homomorphism reductions on Polish groupshttps://zbmath.org/1522.031882023-12-07T16:00:11.105023Z"Beros, Konstantinos A."https://zbmath.org/authors/?q=ai:beros.konstantinos-aSummary: In an earlier paper, we introduced the following pre-order on the subgroups of a given Polish group: if \(G\) is a Polish group and \(H,L \subseteq G\) are subgroups, we say \(H\) is homomorphism reducible to \(L\) iff there is a continuous group homomorphism \(\phi : G \rightarrow G\) such that \(H = \phi ^{-1} (L)\). We previously showed that there is a \(K_\sigma \) subgroup \(L\) of the countable power of any locally compact Polish group \(G\) such that every \(K_\sigma \) subgroup of \(G^\omega \) is homomorphism reducible to \(L\). In the present work, we show that this fails in the countable power of the group of increasing homeomorphisms of the unit interval.On isometry and isometric embeddability between ultrametric Polish spaceshttps://zbmath.org/1522.031912023-12-07T16:00:11.105023Z"Camerlo, Riccardo"https://zbmath.org/authors/?q=ai:camerlo.riccardo"Marcone, Alberto"https://zbmath.org/authors/?q=ai:marcone.alberto"Motto Ros, Luca"https://zbmath.org/authors/?q=ai:motto-ros.lucaSummary: We study the complexity with respect to Borel reducibility of the relations of isometry and isometric embeddability between ultrametric Polish spaces for which a set \(D\) of possible distances is fixed in advance. These are, respectively, an analytic equivalence relation and an analytic quasi-order and we show that their complexity depends only on the order type of \(D\). When \(D\) contains a decreasing sequence, isometry is Borel bireducible with countable graph isomorphism and isometric embeddability has maximal complexity among analytic quasi-orders. If \(D\) is well-ordered the situation is more complex: for isometry we have an increasing sequence of Borel equivalence relations of length \(\omega_1\) which are cofinal among Borel equivalence relations classifiable by countable structures, while for isometric embeddability we have an increasing sequence of analytic quasi-orders of length at least \(\omega + 3\).
We then apply our results to solve various open problems in the literature. For instance, we answer a long-standing question of \textit{S. Gao} and \textit{A. S. Kechris} [On the classification of Polish metric spaces up to isometry. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1012.54038)] by showing that the relation of isometry on locally compact ultrametric Polish spaces is Borel bireducible with countable graph isomorphism.The descriptive complexity of the set of all closed zero-dimensional subsets of a Polish spacehttps://zbmath.org/1522.031922023-12-07T16:00:11.105023Z"Debs, Gabriel"https://zbmath.org/authors/?q=ai:debs.gabriel"Saint Raymond, Jean"https://zbmath.org/authors/?q=ai:saint-raymond.jeanSummary: Given a space \(X\) we investigate the descriptive complexity class \(\Gamma_X\) of the set \(\mathcal{F}_0(X)\) of all its closed zero-dimensional subsets, viewed as a subset of the hyperspace \(\mathcal{F}(X)\) of all closed subsets of \(X\). We prove that \(\max \{\Gamma_X; X \text{ analytic } \} = \mathbf{\Sigma}_2^1\) and \(\sup\{\Gamma_X; X \text{ Borel } \mathbf{\Pi}_\xi^0\} \supseteq \Game \mathbf{\Sigma}_\xi^0\) for any countable ordinal \(\xi \geq 1\). In particular we prove that there exists a one-dimensional Polish subpace of \(2^\omega \times \mathbb{R}^2\) for which \(\mathcal{F}_0(X)\) is not in the smallest non trivial pointclass closed under complementation and the Souslin operation \(\mathcal{A}\).The Dyck and the Preiss separation uniformlyhttps://zbmath.org/1522.031932023-12-07T16:00:11.105023Z"Gregoriades, Vassilios"https://zbmath.org/authors/?q=ai:gregoriades.vassiliosSummary: We are concerned with two separation theorems about analytic sets by Dyck and Preiss, the former involves the positively-defined subsets of the Cantor space and the latter the Borel-convex subsets of finite dimensional Banach spaces. We show by introducing the corresponding separation trees that both of these results admit a constructive proof. This enables us to give the uniform version of these separation theorems, and to derive as corollaries the results, which are analogous to the fundamental fact ``\(\mathsf{HYP}\) is effectively bi-analytic'' provided by the Suslin-Kleene Theorem.Borel functions and separability of metric spaceshttps://zbmath.org/1522.031942023-12-07T16:00:11.105023Z"Gu, Kai"https://zbmath.org/authors/?q=ai:gu.kaiSummary: Let \(X\) and \(Y\) be metric spaces with \(X\) separable, and let \(f: X\rightarrow Y\) be a Borel function. Is \(f(X)\) then separable? In this paper, we prove that this problem is independent of ZFC. We also give a partial answer to an open problem which was asked by \textit{A. H. Stone} [Lect. Notes Math. 375, 242--248 (1974; Zbl 0288.28006)].Games orbits play and obstructions to Borel reducibilityhttps://zbmath.org/1522.031962023-12-07T16:00:11.105023Z"Lupini, Martino"https://zbmath.org/authors/?q=ai:lupini.martino"Panagiotopoulos, Aristotelis"https://zbmath.org/authors/?q=ai:panagiotopoulos.aristotelisSummary: We introduce a new, game-theoretic approach to anti-classification results for orbit equivalence relations. Within this framework, we give a short conceptual proof of Hjorth's turbulence theorem. We also introduce a new dynamical criterion providing an obstruction to classification by orbits of CLI groups. We apply this criterion to the relation of equality of countable sets of reals, and the relations of unitary conjugacy of unitary and selfadjoint operators on the separable infinite-dimensional Hilbert space.The class of non-Desarguesian projective planes is Borel completehttps://zbmath.org/1522.031972023-12-07T16:00:11.105023Z"Paolini, Gianluca"https://zbmath.org/authors/?q=ai:paolini.gianlucaSummary: For every infinite graph \(\Gamma\) we construct a non-Desarguesian projective plane \(P^*_{\Gamma}\) of the same size as \( \Gamma \) such that \( \mathrm{Aut}(\Gamma ) \cong \mathrm{Aut}(P^*_{\Gamma })\) and \( \Gamma _1 \cong \Gamma _2\) iff \( P^*_{\Gamma _1} \cong P^*_{\Gamma _2}\). Furthermore, restricted to structures with domain \( \omega \), the map \( \Gamma \mapsto P^*_{\Gamma }\) is Borel. On one side, this shows that the class of countable non-Desarguesian projective planes is Borel complete, and thus not admitting an Ulm type system of invariants. On the other side, we rediscover the main result of \textit{E. Mendelsohn} [J. Geom. 2, 97--106 (1972; Zbl 0235.50013)] on the realizability of every group as the group of collineations of some projective plane. Finally, we use classical results of projective geometry to prove that the class of countable Pappian projective planes is Borel complete.A characterization of \(\mathbf {\Sigma}_{2}^{1}\) setshttps://zbmath.org/1522.031982023-12-07T16:00:11.105023Z"Pawlikowski, Janusz"https://zbmath.org/authors/?q=ai:pawlikowski.januszSummary: We show that a subset \(X\) of a given Polish space \(\mathcal X\) is \(\mathbf{\Sigma}_{2}^{1}\) iff there is an open set \(O\subseteq\mathcal X\times[\omega]^{\omega}\) such that
\[
X=\{x\in\mathcal X\colon\exists r\in[\omega]^{\omega}\;\{x\}\times[r]^{\omega}\subseteq O\}.
\]
This implies that if a set \(U\subseteq\omega^{\omega}\times(\mathcal X\times[\omega]^{\omega})\) is universal for \(G_{\delta}\) subsets of \(\mathcal X\times[\omega]^{\omega}\), then the set of all \((v,x)\in\omega^{\omega}\times\mathcal X\) such that the section \(U_{vx}\) has nonempty interior in the Ellentuck topology is universal for \(\mathbf{\Sigma}_{2}^{1}\) subsets of \(\mathcal X\). It follows that the \(\sigma\)-ideal of meager sets in the Ellentuck topology is not \(\mathbf{\Sigma}_{2}^{1}\) on \(G_{\delta}\), a fact established recently by \textit{M. Sabok} [Adv. Math. 230, No. 3, 1184--1195 (2012; Zbl 1259.03062)] with the help of Kleene's Recursion Theorem.On Borel maps, calibrated \(\sigma\)-ideals, and homogeneityhttps://zbmath.org/1522.031992023-12-07T16:00:11.105023Z"Pol, R."https://zbmath.org/authors/?q=ai:pol.roman"Zakrzewski, P."https://zbmath.org/authors/?q=ai:zakrzewski.piotrSummary: Let \( \mu \) be a Borel measure on a compactum \( X\). The main objects in this paper are \( \sigma\)-ideals \( I(\dim )\), \( J_0(\mu )\), \( J_f(\mu )\) of Borel sets in \( X\) that can be covered by countably many compacta which are finite-dimensional, or of \( \mu \)-measure null, or of finite \( \mu \)-measure, respectively. Answering a question of \textit{J. Zapletal} [Topology Appl. 167, 31--35 (2014; Zbl 1349.03057)], we shall show that for the Hilbert cube, the \( \sigma\)-ideal \( I(\dim )\) is not homogeneous in a strong way. We shall also show that in some natural instances of measures \( \mu \) with nonhomogeneous \( \sigma\)-ideals \( J_0(\mu )\) or \( J_f(\mu )\), the completions of the quotient Boolean algebras \(\mathrm{Borel}(X)/J_0(\mu )\) or \(\mathrm{Borel}(X)/J_f(\mu )\) may be homogeneous.
We discuss the topic in a more general setting, involving calibrated \( \sigma\)-ideals.On Katětov and Katětov-Blass orders on analytic P-ideals and Borel idealshttps://zbmath.org/1522.032002023-12-07T16:00:11.105023Z"Sakai, Hiroshi"https://zbmath.org/authors/?q=ai:sakai.hiroshi.1Summary: \textit{H. Minami} and \textit{H. Sakai} [Arch. Math. Logic 55, No. 7--8, 883--898 (2016; Zbl 1362.03045)] investigated the cofinal types of the Katětov and the Katětov-Blass orders on the family of all \(F_\sigma \) ideals. In this paper we discuss these orders on analytic P-ideals and Borel ideals. We prove the following:{\parindent=7mm\begin{itemize}\item[--]The family of all analytic P-ideals has the largest element with respect to the Katětov and the Katětov-Blass orders.\item[--]The family of all Borel ideals is countably upward directed with respect to the Katětov and the Katětov-Blass orders.
\end{itemize}} In the course of the proof of the latter result, we also prove that for any analytic ideal \(\mathcal {I}\) there is a Borel ideal \(\mathcal {J}\) with \(\mathcal {I} \subseteq \mathcal {J}\).Continuous reducibility and dimension of metric spaceshttps://zbmath.org/1522.032012023-12-07T16:00:11.105023Z"Schlicht, Philipp"https://zbmath.org/authors/?q=ai:schlicht.philippSummary: If \((X,d)\) is a Polish metric space of dimension 0, then by Wadge's lemma, no more than two Borel subsets of \(X\) are incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space \((X,d)\) of positive dimension, there are uncountably many Borel subsets of \((X,d)\) that are pairwise incomparable with respect to continuous reducibility. In general, the reducibility that is given by the collection of continuous functions on a topological space \((X,\tau )\) is called the \textit{Wadge quasi-order} for \((X,\tau )\). As an application of the main result, we show that this quasi-order, restricted to the Borel subsets of a Polish space \((X,\tau )\), is a \textit{well-quasiorder} if and only if \((X,\tau )\) has dimension 0. Moreover, we give further examples of applications of the construction of graph colorings that is used in the proofs.The linear refinement number and selection theoryhttps://zbmath.org/1522.032042023-12-07T16:00:11.105023Z"Machura, Michał"https://zbmath.org/authors/?q=ai:machura.michal"Shelah, Saharon"https://zbmath.org/authors/?q=ai:shelah.saharon"Tsaban, Boaz"https://zbmath.org/authors/?q=ai:tsaban.boazSummary: The \textit{linear refinement number} \(\mathfrak {lr}\) is the minimal cardinality of a centered family in \({[\omega]^{\omega}}\) such that no linearly ordered set in \(({[\omega]^{\omega}},\subseteq ^*)\) refines this family. The \textit{linear excluded middle number} \(\mathfrak {lx}\) is a variation of \(\mathfrak {lr}\). We show that these numbers estimate the critical cardinalities of a number of selective covering properties. We compare these numbers to the classical combinatorial cardinal characteristics of the continuum. We prove that \(\mathfrak {lr}=\mathfrak {lx}=\mathfrak {d}\) in all models where the continuum is at most \(\aleph_2\), and that the cofinality of \(\mathfrak {lr}\) is uncountable. Using the method of forcing, we show that \(\mathfrak {lr}\) and \(\mathfrak {lx}\) are not provably equal to \(\mathfrak {d}\), and rule out several potential bounds on these numbers. Our results solve a number of open problems.A parametrized diamond principle and union ultrafiltershttps://zbmath.org/1522.032122023-12-07T16:00:11.105023Z"Fernández-Bretón, David"https://zbmath.org/authors/?q=ai:fernandez-breton.david-j"Hrušák, Michael"https://zbmath.org/authors/?q=ai:hrusak.michaelSummary: We consider a cardinal invariant closely related to Hindman's theorem. We prove that this invariant is small in the iterated Sacks perfect set forcing model, and its corresponding parametrized diamond principle implies the existence of union ultrafilters. As a corollary, we obtain the existence of union ultrafilters in the iterated Sacks model of set theory.Products of Menger spaces in the Miller modelhttps://zbmath.org/1522.032222023-12-07T16:00:11.105023Z"Zdomskyy, Lyubomyr"https://zbmath.org/authors/?q=ai:zdomskyy.lyubomyrSummary: We prove that in the Miller model the Menger property is preserved by finite products of metrizable spaces. This answers several open questions and gives another instance of the interplay between classical forcing posets with fusion and combinatorial covering properties in topology.A long chain of P-pointshttps://zbmath.org/1522.032282023-12-07T16:00:11.105023Z"Kuzeljevic, Borisa"https://zbmath.org/authors/?q=ai:kuzeljevic.borisa"Raghavan, Dilip"https://zbmath.org/authors/?q=ai:raghavan.dilipSummary: The notion of a \(\delta\)-generic sequence of P-points is introduced in this paper. It is proved assuming the Continuum Hypothesis (CH) that for each \(\delta < \omega_2\), any \(\delta\)-generic sequence of P-points can be extended to an \(\omega_2\)-generic sequence. This shows that the CH implies that there is a chain of P-points of length \(\mathfrak{c}^+\) with respect to both Rudin-Keisler and Tukey reducibility. These results answer an old question of \textit{A. Blass} [Trans. Am. Math. Soc. 179, 145--166 (1973; Zbl 0269.02025)].Aggregation of fuzzy metrics and its application in image segmentationhttps://zbmath.org/1522.032792023-12-07T16:00:11.105023Z"Ralevic, N. M."https://zbmath.org/authors/?q=ai:ralevic.nebojsa-m-ralevic"Delic, M."https://zbmath.org/authors/?q=ai:delic.marija"Nedovic, Lj."https://zbmath.org/authors/?q=ai:nedovic.ljubo-m(no abstract)Graph limits: an alternative approach to \(s\)-graphonshttps://zbmath.org/1522.054212023-12-07T16:00:11.105023Z"Doležal, Martin"https://zbmath.org/authors/?q=ai:dolezal.martinSummary: We show that \(s\)-convergence of graph sequences is equivalent to the convergence of certain compact sets, called shapes, of Borel probability measures. This result is analogous to the characterization of graphon convergence (with respect to the cut distance) by the convergence of envelopes, due to \textit{M. Doležal} et al. J. Comb. Theory, Ser. B 147, 252--298 (2021; Zbl 1458.05177)].
{{\copyright} 2021 Wiley Periodicals LLC}Maximal classes of spaces and domains determined by topologies on function spaces of domainshttps://zbmath.org/1522.060132023-12-07T16:00:11.105023Z"Xi, Xiaoyong"https://zbmath.org/authors/?q=ai:xi.xiaoyong"Liu, Han"https://zbmath.org/authors/?q=ai:liu.han"Ho, Weng Kin"https://zbmath.org/authors/?q=ai:ho.weng-kin"Zhao, Dongsheng"https://zbmath.org/authors/?q=ai:zhao.dongshengSummary: In this paper, inspired by the work of \textit{J. D. Lawson} and \textit{L. Xu} [Appl. Categ. Struct. 11, No. 4, 391--402 (2003; Zbl 1028.54018)], we consider the question of what classes \(\mathcal{A}\) of topological spaces should be paired with classes \(\mathcal{B}\) of domains in order that the Isbell and Scott topologies on function spaces \([A \rightarrow B]\) are identical for \(A \in \mathcal{A}\) and \(B \in \mathcal{B}\). It is shown that: (1) if \(\mathcal{A}\) and \(\mathcal{B}\) are restricted to be a class of core compact spaces and a class of bicomplete domains respectively, a class of core compact and compact spaces is paired with a class of conditionally bounded complete domains; (2) if \(\mathcal{A}\) and \(\mathcal{B}\) are restricted to be a class of core compact and locally connected spaces and a class of bicomplete domains respectively, a class of topological spaces called \textit{RW}-spaces (with finitely many connected components), which was first introduced by \textit{H. Kou} and \textit{M.-K. Luo} [Topology Appl. 129, No. 3, 211--220 (2003; Zbl 1018.06006)] and further studied by Lawson and Xu [loc. cit.] later, is paired with a class of pointed \(L\)-domains (disjoint union of pointed \(L\)-domains). This also gives a solution to a problem posed by Kou and Luo [loc. cit.].Strictly zero-dimensional biframes and a characterisation of congruence frameshttps://zbmath.org/1522.060172023-12-07T16:00:11.105023Z"Manuell, Graham"https://zbmath.org/authors/?q=ai:manuell.graham-rSummary: Strictly zero-dimensional biframes were introduced by \textit{B. Banaschewski} and \textit{G. C. L. Brümmer} [Quaest. Math. 13, No. 3--4, 273--290 (1990; Zbl 0722.54019)] as a class of strongly zero-dimensional biframes including the congruence biframes. We consider the category of strictly zero-dimensional biframes and show it is both complete and cocomplete. We characterise the extremal epimorphisms in this category and explore the special position that congruence biframes hold in it. Finally, we provide an internal characterisation of congruence biframes, and hence, of congruence frames.Ideals of semisimple MV-algebras and convergence along set-theoretic filtershttps://zbmath.org/1522.060232023-12-07T16:00:11.105023Z"Lele, Celestin"https://zbmath.org/authors/?q=ai:lele.celestin"Nganou, Jean B."https://zbmath.org/authors/?q=ai:nganou.jean-bernard"Oumarou, Christian M. S."https://zbmath.org/authors/?q=ai:oumarou.christian-maxime-steveSummary: In this paper, we investigate the connection between ideals of semisimple MV-algebras and set-theoretic filters. In particular, we obtain that there is a bijection between closed ideals (with respect to a specified closure operation) and all filters on some associated set \(X\). Finally, we establish that the space of closed prime ideals is homeomorphic to a well-described space having the Stone-Čech compactification \( \beta X\) as a subspace.Distinguished classes of ideal spaces and their topological propertieshttps://zbmath.org/1522.130172023-12-07T16:00:11.105023Z"Finocchiaro, Carmelo A."https://zbmath.org/authors/?q=ai:finocchiaro.carmelo-antonio"Goswami, Amartya"https://zbmath.org/authors/?q=ai:goswami.amartya"Spirito, Dario"https://zbmath.org/authors/?q=ai:spirito.darioLet \((X,\leq) \) be a partially ordered set. Recall that the coarse lower topology on \(X\) is the topology whose subbasic closed sets are the sets of the type \({x}^\uparrow := \{y \in X \mid x \leq y\}\), for \(x\) varying in \(X\). On the other hand, the coarse lower topology on a partially ordered set \((X, \leq)\) is the coarsest (\texttt{T}\(_0\)-) topology on \(X\) whose the so-called specialization order induced by the topology coincides with the given \(\leq\).
A very natural setting where to apply this general framework is the set Idl\((R)\) of all the ideals of a given commutative ring \(R\), partially ordered by inclusion. Nowadays it is well known that Idl\((R)\), endowed with the coarse lower topology, is a spectral space, that is, it is homeomorphic to the prime spectrum of a ring. Note that the subspace topology on its subset of prime ideals, Spec(\(R)\), induced by the coarse lower topology on Idl\((R)\), is the classical Zariski topology of the prime spectrum of a ring.
In some recent investigations, other new examples of spectral subspaces of Idl\((R)\) have been detected (see, [\textit{C. A. Finocchiaro} et al. [J. Algebra 461, 25--41 (2016; Zbl 1339.13003); \textit{C. A. Finocchiaro} and \textit{D. Spirito}, New York J. Math. 26, 1064--1092 (2020; Zbl 1451.13063)]), by observing that they are closed sets in the constructible topology of the spectral space Idl\((R)\).
The aim of this paper is to find other interesting spectral subspaces of Idl\((R)\) without using the constructible topology. This leads the authors to check expli\-ci\-tly the topological properties of the classical Hochster's criterion for spectrality on distinguished classes of ideals. Among the other things, they also prove the classification of all Noetherian rings for which the space of primary ideals Prm\((R)\) is spectral: (i) the space Prm(R) is spectral; (ii) the space Prm(R) is sober; (iii) \(R\) is a direct product of zero-dimensional rings and of one-dimensional domains.
Reviewer: Marco Fontana (Roma)A Hofmann-Mislove theorem for approach spaceshttps://zbmath.org/1522.180072023-12-07T16:00:11.105023Z"Yu, Junche"https://zbmath.org/authors/?q=ai:yu.junche"Zhang, Dexue"https://zbmath.org/authors/?q=ai:zhang.dexue|zhang.dexue.1Summary: The Hofmann-Mislove theorem says that the ordered set of open filters of the open-set lattice of a sober topological space is isomorphic to the ordered set of compact saturated sets (ordered by reverse inclusion) of that space. This paper concerns a metric analogy of this result. To this end, the notion of compact functions of approach spaces is introduced. Such functions are an analog of compact subsets in the enriched context. It is shown that for a sober approach space \(X\), the metric space of proper open \([0, \infty]\)-filters of the metric space of upper regular functions of \(X\) is isomorphic to the opposite of the metric space of inhabited and saturated compact functions of \(X\), establishing a Hofmann-Mislove theorem for approach spaces.On saturated prefilter monadshttps://zbmath.org/1522.180102023-12-07T16:00:11.105023Z"Zhang, Gao"https://zbmath.org/authors/?q=ai:zhang.gao"He, Wei"https://zbmath.org/authors/?q=ai:he.wei.2|he.wei.3|he.wei.1|he.weiSummary: In this paper we show that the prime saturated prefilter monads are sup-dense and interpolating in saturated prefilter monads. It follows that CNS spaces are the lax algebras for prime saturated prefilter monads. As for the algebraic part, we prove that the Eilenberg-Moore algebras for saturated prefilter monads are exactly continuous \(I\)-lattices.On presheaf submonads of quantale-enriched categorieshttps://zbmath.org/1522.180122023-12-07T16:00:11.105023Z"Clementino, Maria Manuel"https://zbmath.org/authors/?q=ai:clementino.maria-manuel"Fitas, Carlos"https://zbmath.org/authors/?q=ai:fitas.carlosThe authors' abstract says: ``This paper focuses on the presheaf monad, or the free cocompletion monad, and its submonads on the realm of \(V\)-categories, for a quantale \(V\). First we present two characterisations of presheaf submonads, both using \(V\)-distributors: one based on admissible classes of \(V\)-distributors, and other using Beck-Chevalley conditions on \(V\)-distributors. Further we prove that lax idempotency for 2-monads on \(V\)-Cat can be characterized via such a Beck-Chevalley condition. Then we focus on the study of the Eilenberg-Moore categories of algebras for our monads, having as main examples the formal ball monad and the Lawvere-Cauchy completion monad.''
Here \(V\) is a complete lattice equipped with a commutative monoid structure such that the monoid operation, written as \(\otimes\), is distributive with respect to arbitrary joins; following the quantale theory terminology, the authors also say that \(V\) is a commutative unital quantale, and briefly call it just a quantale. The paper substantially contributes to the theory of quantale enriched categories, whose examples include the category of (pre)ordered sets and variations of the category of metric spaces among many others.
Reviewer: George Janelidze (Cape Town)Hurwitz-Ran spaceshttps://zbmath.org/1522.180172023-12-07T16:00:11.105023Z"Bianchi, Andrea"https://zbmath.org/authors/?q=ai:bianchi.andrea.1|bianchi.andreaIn a previous paper [``Partially multiplicative quandles and simplicial Hurwitz spaces'', Preprint, \url{arXiv:2106.09425}], the author introduced the notion of \textit{partially multiplicative quandle} (PMQ) and for a PMQ \(\mathcal{Q}\) he defined a \textit{simplicial Hurwitz space} \(\mathrm{Hur}^{\Delta}(\mathcal{Q})\), which is the difference of the geometric realizations of two bisimplicial sets, being equipped with a cell stratification. The construction requires \(\mathcal{Q}\) to be \textit{augmented} \ as a PMQ. A point in \(\mathrm{Hur}^{\Delta}(\mathcal{Q})\) can be regarded as the datum of a finite subset \(P\) of the open unit space \((0,1)^{2}\subset\mathbb{C}\), together with a \textit{monodromy} \(\psi\), defined on certain loops of \(\mathbb{C}\backslash P\) and taking values in \(\mathcal{Q}\).
This paper introduces, for a \textit{semi-algebraic} subspace \(\mathcal{X}\subset\mathbb{H}\) of the closed upper half-plane \(\mathbb{H}=\left\{ \mathfrak{R}\geq0\right\} \subset\mathbb{C}\), and for a PMQ \(\mathcal{Q}\), a \textit{Hurwitz-Ran space} \(\mathrm{Hur}(\mathcal{X};\mathcal{Q})\). More generally, the author introduces, for a \textit{nice couple} \((\mathcal{X},\mathcal{Y})\) of subspaces \(\mathcal{Y}\subseteq\mathcal{X}\) of \(\mathbb{H}\) and for a PMQ-group pair \((\mathcal{Q},G)\), a Hurwitz-Ran space \(\mathrm{Hur}(\mathcal{X},\mathcal{Y};\mathcal{Q},G)\), which can be thought of as a relative version of \(\mathrm{Hur}(\mathcal{X};\mathcal{Q})\).
The synopsis of the paper goes as follows.
\begin{itemize}
\item[\S 2] introduces the notion of nice couple \(\mathfrak{C}=(\mathcal{X},\mathcal{Y})\) of subspaces \(\mathcal{Y}\subset \mathcal{X}\) of the closed upper half plane \(\mathbb{H}\). For each finite subset \(P\subset\mathcal{X}\), the author introduces the \textit{fundamental PMQ} \(\mathfrak{D}_{\mathfrak{C}}(P)\), arising as a subset of \(\pi_{1}(\mathbb{C}\backslash P)\), which allows of defining configurations in \(\mathrm{Hur}(\mathfrak{C};\mathcal{Q},G)\) supported on the set \(P\). The author also introduces the notion of \textit{covering} \underline{\(U\)} of a finite subset \(P\subset\mathcal{X}\), associating several PMQs also with the datum of a finite set and a covering of it.
\item[\S 3] defines the Hurwitz-Ran space \(\mathrm{Hur}(\mathfrak{C};\mathcal{Q},G)\) for each nicee couple \(\mathfrak{C}\) and each PMQ-group pair \((\mathcal{Q},G)\), first as a set and then as a Hausdorff topological space.
\item[\S 4] introduces \textit{morphisms} and \textit{lax morphisms} of nice couples, establishing
Theorem. The Hurwitz-Ran space \(\mathrm{Hur}(\mathcal{X},\mathcal{Y};\mathcal{Q},G)\) are functorial both in the nice couple \((\mathcal{X},\mathcal{Y})\), with respect to morphisms of nice couples, and in the PMQ-group pair. If we restrict to complete PMQs, then functoriality holds also with respect to lax morphisms of nice couples.
\item[\S 5] gives some applications of functoriality of Hurwitz-Ran spaces, particularly studying some local properties of their topology.
\item[\S 6] introduces the \textit{total monodromy}, which is a discrete, continuous invariant of configurations in Hurwitz-Ran spaces. The total monodromy is a map
\[
\omega:\mathrm{Hur}(\mathfrak{C};\mathcal{Q},G)\rightarrow G
\]
in the relative case, and
\[
\widehat{\omega}:\mathrm{Hur}(\mathcal{X};\mathcal{Q})\rightarrow\widehat{\mathcal{Q}}
\]
in the absolute case, where \(\widehat{\mathcal{Q}}\) is the completion of \(\mathcal{Q}\). Three actions of \(G\) on Hurwitz-Ran spaces\ are defined.
\item[\S 7] introduces, in the hypothesis that \(\mathcal{Q}\) is augmented, a subspace \(\mathrm{Hur}(\mathfrak{C};\mathcal{Q}_{+},G)\) of \(\mathrm{Hur}(\mathfrak{C};\mathcal{Q},G)\). Using the notion of explosion, it is established that the inclusion
\[
\mathrm{Hur}(\mathfrak{C};\mathcal{Q}_{+},G)\subseteq \mathrm{Hur}(\mathfrak{C};\mathcal{Q},G)
\]
is in several cases a homotopy equivalence.
\item[\S 8] constructs, for an augmented PMQ \(\mathcal{Q}\), a continuous bijection
\[
v:\left\vert \mathrm{Arr}(\mathcal{Q})\right\vert \rightarrow\mathrm{Hur}(\left[ 0,1\right] ^{2};\widehat{\mathcal{Q}}_{+})
\]
where \(\mathrm{Arr}(\mathcal{Q})\) is the bisimplicial set from [arXiv:2106.09425], Definition 6.6]. It is established that
Theorem. If \(\mathcal{Q}\) is augmented and locally finite, then the simplicial Hurwitz space \(\mathrm{Hur}^{\Delta}(\mathcal{Q})\) is homeomorphic to \(\mathrm{Hur}((0,1)^{2};\mathcal{Q}_{+})\).
\item[\S 9] proves that
\[
v:\mathrm{Hur}^{\Delta}(\mathcal{Q})\rightarrow\mathrm{Hur} ((0,1)^{2};\mathcal{Q}_{+})
\]
is a homeomorphism under the additional hypothesis that \ is \textit{locally finite} PMQ. The author then turns to \textit{Poincaré}, establishing
Theorem. In order to prove that \(\mathcal{Q}\) is Poincaré (respectively, R-Poincaré), it suffices to check that for all \(a\in\mathcal{Q}\) the corresponding components \(\mathrm{Hur}^{\Delta}(\mathcal{Q})(a)\) of \(\mathrm{Hur}^{\Delta}(\mathcal{Q})\) is a topological manifold (respectively, an R-homology manifold).
\item[Appendix A] deals with the proofs of the most technical lemmas and propositions of the article.
\end{itemize}
Reviewer: Hirokazu Nishimura (Tsukuba)Monoidal closedness of the category of stratified \(L\)-semiuniform convergence spaceshttps://zbmath.org/1522.180182023-12-07T16:00:11.105023Z"Fang, Jinming"https://zbmath.org/authors/?q=ai:fang.jinming"Fang, Zhou"https://zbmath.org/authors/?q=ai:fang.zhouSummary: For an integral, commutative unital quantale \(L\), we show that there exists a tensor product between stratified \(L\)-semiuniform convergence spaces and, with respect to this tensor product, that the category of stratified \(L\)-semiuniform convergence spaces and uniformly continuous maps is a symmetric monoidal category. As a result, we obtain the monoidal closedness of the category of stratified \(L\)-semiuniform convergence spaces and uniformly continuous maps.Extensionality and \(\mathcal{E}\)-connectedness in the category of \(\top\)-convergence spaceshttps://zbmath.org/1522.180192023-12-07T16:00:11.105023Z"Fang, Jinming"https://zbmath.org/authors/?q=ai:fang.jinming"Yue, Yueli"https://zbmath.org/authors/?q=ai:yue.yueliSummary: The extensionality of \(\top\)-convergence spaces is verified for a complete residuated lattice \(L\) with the top element \(\top\). And also the \(\mathcal{E}\)-connectedness of \(\top\)-convergence spaces for a class \(\mathcal{E}\) of \(\top\)-convergence spaces is proposed by generalizing Preuss's connectedness of topological spaces. Then we establish a necessary and sufficient condition that for a class \(\mathcal{K}\) of \(\top\)-convergence spaces, there exists a class \(\mathcal{E}\) of \(\top\)-convergence spaces such that each space of \(\mathcal{K}\) is \(\mathcal{E}\)-connected, where we stress the point that the conclusion benefits from the extensionality of the category of \(\top\)-convergence spaces. We further present a deep relationship between \(\mathcal{E}\)-connectedness and \(T_1\)-separation for \(\top\)-convergence spaces, that is, the \(\mathcal{E}\)-connectedness of each subset in a \(\top\)-convergence space implies that of its closure if and only if \(\mathcal{E}\) precisely is a class of \(\top\)-convergence spaces being \(T_1\)-separated, and as a natural result, the product theorem for \(\mathcal{E}\)-connected \(\top\)-convergence spaces is obtained.Sierpinski object for composite affine spaceshttps://zbmath.org/1522.180202023-12-07T16:00:11.105023Z"Denniston, Jeffrey T."https://zbmath.org/authors/?q=ai:denniston.jeffrey-t"Solovyov, Sergey A."https://zbmath.org/authors/?q=ai:solovyov.sergey-aSummary: Motivated by the concept of Sierpinski object for the category of affine bitopological spaces (two topologies instead of one) of \textit{R. Noor} et al. [Iran. J. Fuzzy Syst. 15, No. 3, 65--77 (2018; Zbl 1417.54014)], we construct a functor from the category of affine spaces to the category of composite affine spaces (a set-indexed family of topologies instead of one topology), and show a simple condition, under which this functor preserves Sierpinski object. Thus, we get a convenient method of obtaining a composite affine Sierpinski space, given an affine Sierpinski space.Prekernels of topologically axiomatized subcategories of concrete categorieshttps://zbmath.org/1522.180222023-12-07T16:00:11.105023Z"Infusino, F."https://zbmath.org/authors/?q=ai:infusino.federico-gLet \(\mathcal C\) be a concrete category over a fixed category \(\mathcal X\), i.e. there exits a functor \(U_{\mathcal C}:{\mathcal C}\to {\mathcal X}\). The author introduces the notion of Moore-concrete subcategory. A full subcagory \({\mathcal B}\subseteq {\mathcal C}\) is a Moore-concrete subcategory if for each \(X\in {\mathcal X}\) we have that \(U_{\mathcal B}^{-1}(X)\) is a lower semicomplete lower subsemilattice of \(U_{\mathcal C}^{-1}(X)\) and it contains the top element of the \(\mathcal C\)-fiber of \(X\). In Section 3, the author studies the properties of this class of subcategories. In particular, it is shown that proper Moore-concrete subcategories of \(\mathcal C\) coincides with the reflective modification of \(\mathcal C\). Next section is devoted to the study of prekernels (see [\textit{A. Facchini} and \textit{C. A. Finocchiaro}, Ann. Mat. Pura Appl. (4) 199, No. 3, 1073--1089 (2020; Zbl 1481.18002)]) and precokernels of proper Moore-concrete subcategories. In the last Section, prekernels and precokernels in functor-structurd categories are considered. A correspondence between the \({\mathcal Z}\)-prekernels (resp. \({\mathcal Z}\)-precokernels) of \({\mathcal B}^*\) (i.e. the full subcategory of \(\mathcal B\) obtained by deleting the objects with empty group set) and the kernel (resp. weak cokernels) of a suitable stable category \(({\mathcal B}/{\mathcal R})^*\) obtaining from a congruence \(\mathcal R\) is given. This result generalizes the preorder categories considered in the aforementioned mentioned paper.
Reviewer: Blas Torrecillas (Almería)Free group of Hamel functionshttps://zbmath.org/1522.200922023-12-07T16:00:11.105023Z"Lichman, Mateusz"https://zbmath.org/authors/?q=ai:lichman.mateusz"Pawlikowski, Michał"https://zbmath.org/authors/?q=ai:pawlikowski.michal"Smolarek, Szymon"https://zbmath.org/authors/?q=ai:smolarek.szymon"Swaczyna, Jarosław"https://zbmath.org/authors/?q=ai:swaczyna.jaroslawSummary: We construct a free group of continuum many generators among those autobijections of \(\mathbb{R}\) which are also Hamel bases of \(\mathbb{R}^2\), with the identity function included. We also find two new cases when a real function is a composition of two real functions which are Hamel bases of \(\mathbb{R}^2\).
For the entire collection see [Zbl 1518.26001].Combinatorially rich sets in arbitrary semigroupshttps://zbmath.org/1522.202212023-12-07T16:00:11.105023Z"Hindman, Neil"https://zbmath.org/authors/?q=ai:hindman.neil"Hosseini, Hedie"https://zbmath.org/authors/?q=ai:hosseini.hedie"Strauss, Dona"https://zbmath.org/authors/?q=ai:strauss.dona"Tootkaboni, M. A."https://zbmath.org/authors/?q=ai:tootkaboni.mohammad-akbariIn this paper, the authors establish the definition of combinatorially rich sets in arbitrary semigroups as initiated by \textit{V. Bergelson} and \textit{D. Glasscock} [J. Comb. Theory, Ser. A 172, Article ID 105203, 60 p. (2020; Zbl 1433.05313)] to arbitrary semigroups. Further, the relationship of such type of sets to other notions of largeness in semigroups is studied, and the methodologies employed are powerful. The paper also contains an updated and resourceful reference list at the end.
Reviewer: Sanjib Kumar Datta (Kalyani)Correction to: ``Expansive actions of automorphisms of locally compact groups \(G\) on \(\mathrm{Sub}_{G}\)''https://zbmath.org/1522.220012023-12-07T16:00:11.105023Z"Prajapati, Manoj B."https://zbmath.org/authors/?q=ai:prajapati.manoj-b"Shah, Riddhi"https://zbmath.org/authors/?q=ai:shah.riddhiCorrection to the authors' paper [ibid. 193, No. 1, 129--142 (2020; Zbl 1472.22005)].Iterated function system of generalized cyclic contractions in partial metric spaceshttps://zbmath.org/1522.280092023-12-07T16:00:11.105023Z"Makhoshi, Vuledzani"https://zbmath.org/authors/?q=ai:makhoshi.vuledzani"Khumalo, Melusi"https://zbmath.org/authors/?q=ai:khumalo.melusi"Nazir, Talat"https://zbmath.org/authors/?q=ai:nazir.talatSummary: We generate a fractal using a finite collection of generalized cyclic contraction mappings, belonging to a particular category of mappings defined on a partial metric space. As a consequence, different results are attained for iterated function system that satisfy a different set of generalized cyclic contraction conditions. To substantiate the proven results, an example together with some applications are presented. With these results, we extend, unify and generalize some common results in contemporary literature.Expansive properties of induced dynamical systemshttps://zbmath.org/1522.370142023-12-07T16:00:11.105023Z"Jardón, Daniel"https://zbmath.org/authors/?q=ai:jardon.daniel"Sánchez, Iván"https://zbmath.org/authors/?q=ai:sanchez.ivanSummary: For a given metric space \(X\), we consider the set of all normal fuzzy sets on \(X\), denoted by \(\mathcal{F}(X)\). In this work, we study expanding, positively expansive and weakly positively expansive dynamical systems \((X, f)\) and how they are reflected in the dynamical system \((\mathcal{F}(X), \hat{f})\), where \(\hat{f}\) is the Zadeh's extension of \(f\) and \(\mathcal{F}(X)\) has one of the following metrics: the levelwise metric, the endograph metric, the sendograph metric and the Skorokhod metric. We mainly show that if we consider the following conditions:
\begin{itemize}
\item \((X, f)\) is positively expansive (resp. expanding);\item \((\mathcal{K}(X), \overline{f})\) is positively expansive (resp. expanding);
\item \(( \mathcal{F}_\infty(X), \hat{f})\) is positively expansive (resp. expanding);
\item \(( \mathcal{F}_0(X), \hat{f})\) is positively expansive (resp. expanding).
\end{itemize}
Then (iv)\(\Rightarrow\) (iii) \(\Leftrightarrow\) (ii) \(\Rightarrow\) (i). For expanding dynamical systems, we present a compact metric space and a locally compact metric space to show that (i) \(\nRightarrow\) (ii) and (iii) \(\nRightarrow\) (iv), respectively. For positively expansive dynamical systems, there is a compact metric space satisfying that (i) \(\nRightarrow\) (ii), but we don't know if (iii) \(\Rightarrow\) (iv).Force recurrence near zerohttps://zbmath.org/1522.370202023-12-07T16:00:11.105023Z"Singha, Manoranjan"https://zbmath.org/authors/?q=ai:singha.manoranjan"Hom, Ujjal Kumar"https://zbmath.org/authors/?q=ai:hom.ujjal-kumarSummary: Sets that enjoy with recurrence properties near zero, in particular, sets that force recurrence near zero, sets that contain broken IP-set near zero, sets that force uniform recurrence near zero and sets that contain broken syndetic set near zero have been introduced and some relations among them are established on countable dense subsemigroup of the additive semigroup of positive real numbers. The article begins with a characterization of recurrent point near zero that affects the results exhibited here. It is also showed that how IP-set near zero can be realized dynamically.Topological stability for fuzzy expansive mapshttps://zbmath.org/1522.370212023-12-07T16:00:11.105023Z"Badilla, L."https://zbmath.org/authors/?q=ai:badilla.lautaro"Carrasco-Olivera, D."https://zbmath.org/authors/?q=ai:carrasco-olivera.dante"Sirvent, V. F."https://zbmath.org/authors/?q=ai:sirvent.victor-f"Villavicencio, H."https://zbmath.org/authors/?q=ai:villavicencio.helmuthSummary: We introduce the definitions of expansivity and topological stability for homeomorphisms on fuzzy metric spaces. We show some basic properties of fuzzy expansive homeomorphisms. Moreover we prove Walters's theorem in the context of fuzzy metric spaces, i.e., a fuzzy expansive system with the fuzzy shadowing property is fuzzy topologically stable.On two notions of fuzzy topological entropyhttps://zbmath.org/1522.370232023-12-07T16:00:11.105023Z"Cánovas, Jose S."https://zbmath.org/authors/?q=ai:canovas.jose-s"Kupka, Jiří"https://zbmath.org/authors/?q=ai:kupka.jiriSummary: We explore the notion of fuzzy topological entropy when different definitions of fuzzy compactness are considered. We prove that the recent definitions by \textit{I. Tok} [Trans. Acad. Sci. Azerb., Ser. Phys.-Tech. Math. Sci. 19, No. 1--2, Math. Mech., 162--169, 269 (1999; Zbl 1181.37014)] and by \textit{B. M. U. Afsan} and \textit{C. K. Basu} [Appl. Math. Lett. 24, No. 12, 2030--2033 (2011; Zbl 1269.54003)] are not the most appropriate since they always display zero entropy. Furthermore, we give a simple proof of the bridge result, which states that topological entropy agrees with fuzzy topological entropy when it is defined by using Lowen's definition of compactness. Consequently, many natural properties of the fuzzy topological entropy (such as monotonicity) are obtained as direct corollaries. The particular case of interval maps is also briefly discussed.Chaos and weak mixing on uniform spaceshttps://zbmath.org/1522.370302023-12-07T16:00:11.105023Z"Shao, Hua"https://zbmath.org/authors/?q=ai:shao.huaSummary: Let \(f_{0, \infty} = \{f_n\}_{n = 0}^\infty\) be a sequence of uniformly continuous self-maps on a uniform space \(X\). We prove that under some natural conditions, topological weak mixing implies dense distributional \(\eta\)-chaos in a sequence, and generic \(\eta\)-chaos is equivalent to dense \(\eta^\prime\)-chaos for \((X, f_{0, \infty})\). We give several equivalent characterizations of sensitivity for \((X, f_{0, \infty})\), and as applications, we get the relationships between sensitivity and Li-Yorke sensitivity, and further obtain that topological weak mixing implies Li-Yorke sensitivity for a class of abelian \((X, f_{0, \infty})\). We show that \((X \times Y, f_{0, \infty} \times g_{0, \infty})\) is sensitive (resp. Li-Yorke sensitive and multi-sensitive) if and only if \((X, f_{0, \infty})\) or \((Y, g_{0, \infty})\) is sensitive (resp. Li-Yorke sensitive and multi-sensitive). We also prove that distributional chaos (resp. Li-Yorke chaos, sensitivity and Li-Yorke sensitivity) is equivalent between \((X, f_{0, \infty})\) and its \(N\)-th iteration system. Finally, we confirm that distributional chaos in a sequence, Li-Yorke chaos, sensitivity and Li-Yorke sensitivity are all preserved under topological equi-conjugacy.A note on topological average shadowing property via uniformityhttps://zbmath.org/1522.370312023-12-07T16:00:11.105023Z"Ahmadi, Seyyed Alireza"https://zbmath.org/authors/?q=ai:ahmadi.seyyed-alireza"Wu, Xinxing"https://zbmath.org/authors/?q=ai:wu.xinxingThe authors study the topological ergodic shadowing property and the topological average shadowing property for a continuous map on a uniform space. More precisely, let \(f\) be a continuous map on a uniform compact space \(X\). The authors prove that if \(f\) has the topological ergodic shadowing property, then it has the topological average shadowing property. The result is applied to show that if a uniform continuous surjective map \(f\) has the topological shadowing property, then the following properties are equivalent:
(1) Topological \(\underline{d}\)-shadowing property;
(2) Topological ergodic shadowing property;
(3) Topological average shadowing property.
Moreover, if \(f\) is topological Lyapunov stable and has the topological average shadowing property, then it is topological ergodic.
Reviewer: Ngoc Thach Nguyen (Daejeon)Asymptotic smoothness and universality in Banach spaceshttps://zbmath.org/1522.460062023-12-07T16:00:11.105023Z"Causey, R. M."https://zbmath.org/authors/?q=ai:causey.ryan-michael"Lancien, G."https://zbmath.org/authors/?q=ai:lancien.gillesThe most important contribution in this paper concerns universality of two classes of separable Banach spaces, denoted in this paper by \(A_p\) and \(N_p\) (where \(p\in (1,\infty]\)), which are related to the notion of a Szlenk index.
More concretely, if \(q\) is conjugate to \(p\), then the class \(A_p\) consists of Banach spaces which have \(q\)-summable Szlenk index and the class \(N_p\) consists of Banach spaces \(X\) such that there is \(K>0\) satisfying \(Cz(X,\varepsilon)\leq K\varepsilon^{-q}\), where \(Cz(X,\varepsilon)\) is related to the convex Szlenk index; note that in general we have \(A_p\subset N_p\).
The main result is a construction of \(A_p\) (resp. \(N_p\)) Banach spaces \(\{U_{M,\theta}\colon M\subset \mathbb{N}\) infinite, \(\theta\in(0,1)\}\) with the universal property that for any separable Banach space \(X\in A_p\) (resp. \(X\in N_p\)) there are infinite \(M,M'\subset \mathbb{N}\) and \(\theta,\theta'\in(0,1)\) such that \(X\) is isomorphic to a subspace of \(U_{M,\theta}\) and to a quotient of \(U_{M',\theta'}\). Moreover, the authors prove that it is not possible to find such a family consisting of just one space since, given any \(U\in N_p\), there exists \(X\in A_p\) which is not isomorphic to any subspace of any quotient of \(U\).
Reviewer: Marek Cúth (Praha)Pseudotrees and equivalent norms in the space of continuous functions.https://zbmath.org/1522.460072023-12-07T16:00:11.105023Z"Gul'ko, S. P."https://zbmath.org/authors/?q=ai:gulko.sergey-porfiryevich|gulko.sergei-p"Kobylina, M. S."https://zbmath.org/authors/?q=ai:kobylina.m-sSummary: A class of the pseudotrees is considered. We construct locally compact extension of a pseudotree, which also has the structure of a pseudotree. We prove that the space \(C_0(T)\) of all continuous functions on a locally compact pseudotree \(T\) admits a locally uniform rotund (LUR) renorming if the related space \(C_0(P)\) admits such norm for every subtree \(P\) of \(T\) and an initial segments of \(T\) are separable.The complemented subspace problem for \(C(K)\)-spaces: a counterexamplehttps://zbmath.org/1522.460082023-12-07T16:00:11.105023Z"Plebanek, Grzegorz"https://zbmath.org/authors/?q=ai:plebanek.grzegorz"Salguero-Alarcón, Alberto"https://zbmath.org/authors/?q=ai:salguero-alarcon.albertoA \(C\)-space is a Banach space that is isomorphic to \(C(K)\), the space of all continuous functions on a Hausdorff compact space \(K\).
A longstanding question, originally attributable to \textit{J.~Lindenstrauss} and \textit{D.~E. Wulbert} [J. Funct. Anal. 4, 332--349 (1969; Zbl 0184.15102)], asks whether the class of \(C\)-spaces is closed under taking complemented subspaces. The centerpiece of the paper is the construction of a counterexample to the above problem. More specifically, the authors show that there are two separable (yet non-metrisable) scattered compacta \(K\) and \(L\) with the third Cantor-Bendixson derivative empty, a continuous surjection \(\theta\colon L \to K\) and a closed subspace \(X\) of \(C(L)\) such that
\begin{itemize}
\item[(1)] \(C(L) \simeq \theta^\circ[C(K)] \oplus X\);
\item[(2)] the spaces \( \theta^\circ[C(K)]\) and \(X\) are both \(1\)-complemented in \(C(L)\), where \(\theta^\circ\) is the induced algebra homomorphism;
\item[(3)] the space \(X\) is not a \(C\)-space.
\end{itemize}
From this, the authors derive the existence of a non-\(C\) space \(W\), a compact space \(L\), and a short-exact sequence
\[
0 \to c_0 \to C(L) \to W \to 0.
\]
Reviewer: Tomasz Kania (Praha)Some classical problems of geometric approximation theory in asymmetric spaceshttps://zbmath.org/1522.460122023-12-07T16:00:11.105023Z"Alimov, A. R."https://zbmath.org/authors/?q=ai:alimov.alexey-r"Tsar'kov, I. G."https://zbmath.org/authors/?q=ai:tsarkov.igor-gThe authors continue their investigations on best approximation problems in asymmetric normed spaces. An asymmetric norm on a real vector space \(X\) is a functional \(\|\cdot|:X\to[0,\infty)\) satisfying all the axioms of a norm with positive homogeneity instead of absolute homogeneity (see [\textit{S. Cobzaş}, Functional analysis in asymmetric normed spaces. Basel: Birkhäuser (2013; Zbl 1266.46001)]). For a nonempty subset \(M\) of \(X\) and \(x\in X\), put \(\rho(x,M)=\inf\{\|y-x|: y\in M\}\) (the distance from \(x\) to \(M\)) and \(P_Mx= \{y\in M:\|y-x|=\rho(x,M)\}\) (the metric projection). Due to the asymmetry of the norm \(\|\cdot|\) (that is, the possibility that \(\|{-}x|\ne\|x| \) for some \(x\in X\)), many results from the symmetric case cease to hold in the asymmetric case (e.g., the continuity of the distance function).
The focus in the present paper is on various kinds of solar properties of sets which reveal the structure of approximating sets and, in some cases, can also be useful for designing numerical approximation algorithms.
As the authors point out in the abstract: ``We establish a number of theorems of geometric approximation theory in asymmetrically normed spaces. Sets with continuous selection of the near-best approximation operator are studied and properties of such sets are discussed in terms of \(\delta\)-solar points and the distance function. A result on the coincidence of the classes of \(\delta\)- and \(\gamma\)-suns in asymmetric spaces is given. An asymmetric analogue of the Kolmogorov criterion for an element of best approximation for suns, strict suns, and \(\alpha\)-suns is put forward.''
Reviewer: Stefan Cobzaş (Cluj-Napoca)A characterization of weak proximal normal structure and best proximity pairshttps://zbmath.org/1522.460132023-12-07T16:00:11.105023Z"Digar, Abhik"https://zbmath.org/authors/?q=ai:digar.abhik"Espínola García, Rafael"https://zbmath.org/authors/?q=ai:espinola-garcia.rafael"Kosuru, G. Sankara Raju"https://zbmath.org/authors/?q=ai:kosuru.g-sankara-rajuSummary: The aim of this paper is to address an open problem given in
[\textit{W.~A. Kirk} and \textit{N.~Shahzad}, J. Math. Anal. Appl. 463, No.~2, 461--476 (2018; Zbl 1392.54032)].
We give a characterization of weak proximal normal structure using best proximity pair property. We also introduce a notion of pointwise cyclic contraction with respect to orbits and therein prove the existence of a best proximity pair in the setting of reflexive Banach spaces.A note on Mosco convergence in \(\operatorname{CAT}(0)\) spaceshttps://zbmath.org/1522.460482023-12-07T16:00:11.105023Z"Bërdëllima, A."https://zbmath.org/authors/?q=ai:berdellima.arianSummary: In this note, we show that in a complete \(\mathrm{CAT}(0)\) space pointwise convergence of proximal mappings under a certain normalization condition implies Mosco convergence.A fixed point theorem for a system of Pachpatte operator equationshttps://zbmath.org/1522.470802023-12-07T16:00:11.105023Z"Karapınar, Erdal"https://zbmath.org/authors/?q=ai:karapinar.erdal"Öztürk, Ali"https://zbmath.org/authors/?q=ai:ozturk.ali-ugur"Rakočević, Vladimir"https://zbmath.org/authors/?q=ai:rakocevic.vladimirThe authors study the solvability of the fixed point equation \(Tx= x\), subject to the constraints \(\alpha_1(x)=\cdots = \alpha_r(x)= 0\), where \(T\) and \(\alpha_i\) \((i=1, 2,\dots, r)\) are operators in a Banach space with cone. \textit{A. H. Ansari} et al. [J. Fixed Point Theory Appl. 19, No. 2, 1145--1163 (2017; Zbl 1454.47060)] have proved the existence of a solution, building on a fixed point theorem by \textit{L. B. Čirić} [Publ. Inst. Math., Nouv. Sér. 17(31), 52--58 (1974; Zbl 0309.54035)].
In the present paper, the authors prove existence by means of another fixed point theorem by \textit{B. G. Pachpatte} [Indian J. Pure Appl. Math. 10, 1039--1043 (1979; Zbl 0412.54053)].
Reviewer: Jürgen Appell (Würzburg)A modified Riemannian Halpern algorithm for nonexpansive mappings on Hadamard manifoldshttps://zbmath.org/1522.471112023-12-07T16:00:11.105023Z"Yao, Teng-Teng"https://zbmath.org/authors/?q=ai:yao.tengteng"Li, Ying-Hui"https://zbmath.org/authors/?q=ai:li.yinghui"Zhang, Yong-Shuai"https://zbmath.org/authors/?q=ai:zhang.yongshuai"Zhao, Zhi"https://zbmath.org/authors/?q=ai:zhao.zhiSummary: In this paper, we are concerned with the problem of finding fixed points of nonexpansive mappings on Hadamard manifolds. To solving this kind of problem, a modified Riemannian Halpern algorithm, which is a natural generalization of a modified Halpern algorithm in Euclidean space is proposed. By giving some mild assumptions and necessary lemmas, the global convergence of the proposed algorithm is established. Finally, when the problem is solved in the framework of Hadamard manifolds, the numerical experiments show the effectiveness of the proposed algorithm, especially in computational time and number of iterations.Polynomial growth and asymptotic dimensionhttps://zbmath.org/1522.530302023-12-07T16:00:11.105023Z"Papasoglu, Panos"https://zbmath.org/authors/?q=ai:papasoglu.panosThe author of this paper proves that a graph of polynomial growth strictly less than \(n^{k+1}\) has asymptotic dimension at most \(k\). This result refines the fact shown by \textit{M. Bonamy} et al. [``Asymptotic dimension of minor-closed families and Assouad-Nagata dimension of surfaces'', Preprint, \url{arXiv:2012.02435}] that graphs with polynomial growth have finite asymptotic dimension. As a corollary, it is shown that Riemannian manifolds of bounded geometry with polynomial growth strictly less than \(n^{k+1}\) have asymptotic dimension at most \(k\). Furthermore, the author shows that there are graphs of growth less than \(n^{1+\epsilon}\) for any \(\epsilon >0\) which have infinite asymptotic Assouad-Nagata dimension. Finally, insights and questions are given regarding the relationship between growth functions and coarse embeddings.
Reviewer: Yutaka Iwamoto (Niihama)Topology. Translated from the Italian by Simon G. Chiossihttps://zbmath.org/1522.540012023-12-07T16:00:11.105023Z"Manetti, Marco"https://zbmath.org/authors/?q=ai:manetti.marcoPublisher's description: This is an introductory textbook on general and algebraic topology, aimed at anyone with a basic knowledge of calculus and linear algebra. It provides full proofs and includes many examples and exercises. The covered topics include: set theory and cardinal arithmetic; axiom of choice and Zorn's lemma; topological spaces and continuous functions; con- nectedness and compactness; Alexandrov compactification; quotient topol- ogies; countability and separation axioms; prebasis and Alexander's theorem; the Tychonoff theorem and paracompactness; complete metric spaces and function spaces; Baire spaces; homotopy of maps; the fundamental group; the van Kampen theorem; covering spaces; Brouwer and Borsuk's theorems; free groups and free product of groups and basic category theory. While it is very concrete at the beginning, abstract concepts are gradually introduced.
This second edition contains a new chapter with a topological introduction to sheaf cohomology and applications. It also corrects some inaccuracies and some additional exercises are proposed.
The textbook is suitable for anyone needing a basic, comprehensive introduction to general and algebraic topology and its applications.
See the review of the first edition in [Zbl 1322.54001]. See the review of the original Italian edition in [Zbl 1139.54001]. See the review of the second Italian edition in [Zbl 1307.54001].On the concept of generalization of \(\mathcal{I}\)-density pointshttps://zbmath.org/1522.540022023-12-07T16:00:11.105023Z"Hejduk, Jacek"https://zbmath.org/authors/?q=ai:hejduk.jacek"Wiertelak, Renata"https://zbmath.org/authors/?q=ai:wiertelak.renataThe notion of Lebesgue density point has been defined at the beginning of the XX. century. \textit{W. Poreda} et al. introduced, in [Fundam. Math. 125, 167--173 (1985; Zbl 0613.26002)], the notion of \(\mathcal{I}\)-density point, a category analogue of the notion of Lebesgue density point. Over the least years several modifications of these notions have been introduced and studied by many mathematicians. Two concepts can be mentioned here: \(\mathcal{I}\langle s\rangle\)-density and \(\mathcal{I}(\mathcal{J})\)-density, see e.g. the survey \textit{J. Hejduk} and \textit{R. Wiertelak} [in: Traditional and present-day topics in real analysis. Dedicated to Professor Jan Stanisław Lipiński on the occasion of his 90th birthday. Łódź: Łódź University Press, University of Łódź, Faculty of Mathematics and Computer Science. 431--447 (2013; Zbl 1334.54004)].
In the paper under review, the authors introduce and investigate the following generalization of \(\mathcal{I}\)-density point. Let \(\mathcal{S}\) and \(\mathcal{I}\) denote the \(\sigma\)-algebra of all subsets of \(\mathbb{R}\) having the Baire property, and the ideal of first category subsets of \(\mathbb{R}\). Let \(Y\in \mathcal{S}\setminus\mathcal{I}\) be a bounded set and \((a_n)\), \((b_n)\) be sequences of real numbers converging to \(0\) such that \(a_n\ne 0\) for every \(n\). Set \(Y_n=a_nY+b_n\) for each \(n\in\mathbb{N}\) and \(\mathcal{Y}=(Y_n)\). A point \(x_0\in\mathbb{R}\) is an \textit{\(\mathcal{I}(\mathcal{Y})\)-density point} of a set \(A\in\mathcal{S}\) if for every sequence \(n_k\nearrow\infty\) there exists a subsequence \((n_{k_m})_m\) such that \[\limsup_{m\to\infty} \left(\frac{1}{a_{n_{k_m}}}(Y_{n_{k_m}}\setminus A - x_0 - b_{n_{k_m}})\right )\in\mathcal{I}.\] Now, for every \(A\in\mathcal{S}\) let \(\Phi_{\mathcal{I}(\mathcal{Y})}(A)\) be the set of all \(\mathcal{I}(\mathcal{Y})\)-density points of \(A\). The authors show that the operator \(\mathcal{I}(\mathcal{Y})\) is the lower density operator on the space \((\mathbb{R}, \mathcal{S},\mathcal{I})\), so the family \[\mathcal{T}_{\mathcal{I}(\mathcal{Y})}= \{ A\in\mathcal{S}\colon A\subset\Phi_{\mathcal{I}(\mathcal{Y})}(A)\}\] is a topology on \(\mathbb{R}\).
In the main part of the paper, the authors show that the concept of \(\mathcal{I}(\mathcal{Y})\)-density is an essential extension of the concept of \(\mathcal{I}(\mathcal{J})\)-density and consequently also \(\mathcal{I}\langle s\rangle\)-density and \(\mathcal{I}\)-density.
Finally, they notice that the measure analog of \(\mathcal{I}(\mathcal{Y})\)-density has been considered by \textit{F. Strobin} and \textit{R. Wiertelak} in [Topology Appl. 199, 1--16 (2016; Zbl 1332.54006)].
Reviewer: Tomasz Natkaniec (Gdańsk)Unified theory of the kernel of a set via hereditary classes and generalized topologieshttps://zbmath.org/1522.540032023-12-07T16:00:11.105023Z"Sanabria, José"https://zbmath.org/authors/?q=ai:sanabria.jose-eduardo"Maza, Laura"https://zbmath.org/authors/?q=ai:maza.laura"Rosas, Ennis"https://zbmath.org/authors/?q=ai:rosas.ennis-r"Carpintero, Carlos"https://zbmath.org/authors/?q=ai:carpintero.carlos-rIn this paper the authors investigate a unification of variants of the kernel of a set in a generalized topological space endowed with a hereditary class. Based on this investigation, in the framework of hereditary generalized topological spaces, they introduce and study new modifications of sets and new separation properties.
Reviewer: Dimitrios Georgiou (Pátra)Erdős-Ulam ideals vs. simple density idealshttps://zbmath.org/1522.540042023-12-07T16:00:11.105023Z"Kwela, Adam"https://zbmath.org/authors/?q=ai:kwela.adamSummary: The main aim of this paper is to bridge two directions of research generalizing asymptotic density zero sets. This enables to transfer results concerning one direction to the other one. Consider a function \(g : \omega \rightarrow [0, \infty)\) such that \(\lim_{n \rightarrow \infty} g(n) = \infty\) and \(\frac{n}{g(n)}\) does not converge to 0. Then the family \(\mathcal{Z}_g = \{A \subseteq \omega : \lim_{n \rightarrow \infty} \frac{\text{card}(A \cap n)}{g(n)} = 0 \}\) is an ideal called simple density ideal (or ideal associated to upper density of weight \(g\)). We compare this class of ideals with Erdős-Ulam ideals. In particular, we show that there \(\sqsubseteq\) are antichains of size \(\mathfrak{c}\) among Erdős-Ulam ideals which are and are not simple density ideals (in [\textit{A. Kwela} et al., J. Math. Anal. Appl. 477, No. 1, 551--575 (2019; Zbl 1512.03060)] it is shown that there is also such an antichain among simple density ideals which are not Erdős-Ulam ideals). We characterize simple density ideals which are Erdős-Ulam as those containing the classical ideal of sets of asymptotic density zero. We also characterize Erdős-Ulam ideals which are simple density ideals. In the latter case we need to introduce two new notions. One of them, called increasing-invariance of an ideal \(\mathcal{I}\), asserts that given \(B \in \mathcal{I}\) and \(C \subseteq \omega\) with \(\text{card}(C \cap n) \leq \text{card}(B \cap n)\) for all \(n\), we have \(C \in \mathcal{I}\). This notion is inspired by \textit{M. Balcerzak} et al. [J. Math. Anal. Appl. 445, No. 1, 423--442 (2017; Zbl 1359.40002)] and is later applied in [Kwela et al., loc. cit.] for a partial solution of [\textit{A. Kwela} and \textit{J. Tryba}, Acta Math. Hung. 151, No. 1, 139--161 (2017; Zbl 1399.03010), Problem 5.8]. Finally, we pose some open problems.A new approach to lattice-valued convergence groups via \(\top\)-filtershttps://zbmath.org/1522.540052023-12-07T16:00:11.105023Z"Zhang, Lin"https://zbmath.org/authors/?q=ai:zhang.lin.5"Pang, Bin"https://zbmath.org/authors/?q=ai:pang.binSummary: In this paper, choosing a complete residuated lattice \(L\) as the lattice background, we introduce the concept of \(\top\)-convergence groups, which is a group equipped with a \(\top\)-convergence structure such that the group operations are continuous with respect to the \(\top\)-convergence space. Then we make further investigations on \(\top\)-convergence groups, including (1) We provide many nice properties of \(\top\)-convergence groups and two characterization theorems; (2) Considering \(\top\)-neighborhood groups, we propose topological \(\top\)-convergence groups and establish their relationships with topological \(\top\)-neighborhood groups; (3) We further introduce the notion of \(\top\)-limit groups by equipping a limit condition on \(\top\)-convergence groups and then investigate the uniformizability of \(\top\)-limit groups by means of \(\top\)-uniform limit spaces.Cardinal inequalities with Shanin number and \(\pi\)-characterhttps://zbmath.org/1522.540062023-12-07T16:00:11.105023Z"Gotchev, Ivan"https://zbmath.org/authors/?q=ai:gotchev.ivan-s"Tkachuk, Vladimir"https://zbmath.org/authors/?q=ai:tkachuk.vladimir-vIn this paper the authors present some new ZFC-consistent cardinal inequalities. In particular, they prove that, under GCH, \(|X| \leq sh(X)^{\pi\chi(X)\psi_c(X)}\) (hence \(|X| \leq sh(X)^{\chi(X)}\)) for any Hausdorff space \(X\) and, still under GCH, \(d(X) \leq sh(X)^ {t(X)\pi\chi(X)}\) for every regular space \(X\). Here \(sh(X)= \min\{\kappa \geq \omega: \kappa^+ \hbox{ is a caliber of } X\}\) denotes the Shanin number of \(X\). They also establish in ZFC that \(|X| \leq wL(X)^{\overline{\Delta}(X) 2^{\pi\chi(X)}}\) whenever \(X\) is a Urysohn space. Here \(\overline{\Delta}(X)\) denotes the regular diagonal degree of \(X\). Additionally, some important remarks and examples are given throughout the paper.
Reviewer: Davide Giacopello (Messina)Lindelöf scattered subspaces of nice \(\sigma\)-products are \(\sigma\)-compacthttps://zbmath.org/1522.540072023-12-07T16:00:11.105023Z"Tkachuk, Vladimir V."https://zbmath.org/authors/?q=ai:tkachuk.vladimir-vFor an uncountable set \(A\), consider \(\Sigma_\ast(A) = \{ x \in \mathbb{R}^A : \vert x^{-1}(\mathbb{R}\setminus (-\epsilon,\epsilon)) \vert < \omega, \text{ for any } \epsilon > 0 \}\). Compact subsets of \(\Sigma_\ast(A)\) are called Eberlein compact. A space \(X\) is Lindelöf \(\Sigma\) (or has the Lindelöf \(\Sigma\)-property) if there exists a countable family \(\mathcal{F}\) of subsets of \(X\) such that \(\mathcal{F}\) is a network with respect to a compact cover \(\mathcal{C}\) of the space \(X\).
In the paper under review, by means of an interesting example, the author answers in the negative the question posed in the paper [\textit{V. V. Tkachuk}, Math. Slovaca 67, No. 1, 227--234 (2017; Zbl 1399.54071)] on whether any Lindelöf subspace of an Eberlein compact space must have the Lindelöf \(\Sigma\)-property.
Based on this example, the author proposes a better question, namely whether any scattered Lindelöf subspace of an Eberlein compact space must be \(\sigma\)-compact. Although this question is not answered in the paper, it is proved in one of the main results of the paper that any scattered Lindelöf subspace (a space \(X\) is scattered if every non-empty subspace \(Y\subset X\) has an isolated point) of a \(\sigma\)-product of first countable spaces is \(\sigma\)-compact.
On the other hand, recall that \(Y\) is a \(G_\delta\)-modification of a space \(X\) if \(Y\) is the set \(X\) with the topology generated by all \(G_\delta\)-subsets of \(X\). In this paper, it is established that if \(X\) is the \(G_\delta\)-modification of a scattered compact space, then \(ext(C_p(X)) = \omega\), which gives a positive answer to Questions 4.3 and 4.6 from [loc. cit.] on whether or not the hypothesis that \(X\) is an Eberlein space is necessary.
Since it is also important to know what additional properties the Lindelöf scattered subspaces of Eberlein compact spaces have (in fact, it is not known whether these spaces must be \(\sigma\)-compact), the author closes his paper with a long series of questions, with the aim of making further progress in this area.
Reviewer: Jesús F. Tenorio (Huajuapan de Léon)Many valued topologies on L-setshttps://zbmath.org/1522.540082023-12-07T16:00:11.105023Z"Demirci, Mustafa"https://zbmath.org/authors/?q=ai:demirci.mustafaSummary: Many valued topologies on L-sets, providing a common framework for fuzzy topologies on fuzzy sets and the fixed-basis lattice-valued topologies on ordinary sets, are the main subject of the present study. In this paper, we have introduced many valued topological L-set-spaces, referring to L-sets equipped with many valued topologies, in an axiomatic way and focused on the vindication of their axiomatization in a categorical framework. In order to attain this objective, we apply the theory of Höhle of topological space objects formulated in terms of a partially ordered monad on an abstract category, and show that the category of many valued topological L-set-spaces is isomorphic to the category of topological space objects with respect to a partially ordered monad on a category of L-sets.A novel notion in rough set theory: invariant subspacehttps://zbmath.org/1522.540092023-12-07T16:00:11.105023Z"Gao, Qiang"https://zbmath.org/authors/?q=ai:gao.qiang"Ma, Liwen"https://zbmath.org/authors/?q=ai:ma.liwenSummary: All current work on calculating approximate sets of a rough set inevitably requires the participation of all elements in the universe. However, it is cumbersome and causes a huge waste of resources when the universe is quite big and the computed set is small enough. By introducing the notion of invariant subspace in rough set theory, we skillfully reduce the range of elements involved in the computations of approximations of a rough set from the whole universe \(U\) to a suitable invariant subspace \(V\) of \(U\), and then give the modular method within the reduced range. Moreover, we present the modular Boolean matrix method, such that the calculation of upper and lower approximations of rough sets can be converted into operations on modular matrices. Finally, the results in covering approximation space are generalized to fuzzy \(\beta \)-covering approximation space to calculate the upper and lower approximations of the crisp sets. In particular, an algorithm for calculating the value of \(\beta\) is proposed to make the involved information has the highest distinction degree. This \(\beta\) is more reliable and meaningful than the empirical one, and an example about COVID-19 is put forward to simply illustrate its application.Asymptotic dimension of fuzzy metric spaceshttps://zbmath.org/1522.540102023-12-07T16:00:11.105023Z"Grzegrzolka, Pawel"https://zbmath.org/authors/?q=ai:grzegrzolka.pawelSummary: In this paper, we define asymptotic dimension of fuzzy metric spaces in the sense of \textit{A. George} and \textit{P. Veeramani} [ibid. 64, No. 3, 395--399 (1994; Zbl 0843.54014)]. We prove that asymptotic dimension is an invariant in the coarse category of fuzzy metric spaces. We also show several properties of asymptotic dimension in the fuzzy setting which resemble the properties of asymptotic dimension in the metric setting. We finish with calculating asymptotic dimension of a few fuzzy metric spaces.Some properties of Skorokhod metric on fuzzy setshttps://zbmath.org/1522.540112023-12-07T16:00:11.105023Z"Huang, Huan"https://zbmath.org/authors/?q=ai:huang.huanSummary: This paper discusses the properties of Skorokhod metric on normal and upper semi-continuous fuzzy sets on metric space. All fuzzy sets mentioned below refer to this type of fuzzy sets. We confirm that the Skorokhod metric and the enhanced-type Skorokhod metric are equivalent on compact fuzzy sets. However, the Skorokhod metric and the enhanced-type Skorokhod metric are not necessarily equivalent on \(L_p\)-integrable fuzzy sets, which include compact fuzzy sets. We point out that the \(L_p\)-type \(d_p\) metric, \(p \geq 1\), is well-defined in common cases but the \(d_p\) metric is not always well-defined on all fuzzy sets. We introduce the \(d_p^\ast\) metric which is an expansion of the \(d_p\) metric, and write \(d_p^\ast\) as \(d_p\) in the sequel. Then, we investigate the relationship between these two Skorokhod-type metrics and the \(d_p\) metric. We show that the relationship can be divided into three cases. On compact fuzzy sets, the Skorokhod metric is stronger than the \(d_p\) metric. On \(L_p\)-integrable fuzzy sets, the Skorokhod metric is not necessarily stronger than the \(d_p\) metric, but the enhanced-type Skorokhod metric is still stronger than the \(d_p\) metric. On all fuzzy sets, even the enhanced-type Skorokhod metric is not necessarily stronger than the \(d_p\) metric. We also show that the Skorokhod metric is stronger than the sendograph metric. At last, we give a simple example to answer some recent questions involved the Skorokhod metric.Diagonal conditions and uniformly continuous extension in \(\top\)-uniform limit spaceshttps://zbmath.org/1522.540122023-12-07T16:00:11.105023Z"Jager, G."https://zbmath.org/authors/?q=ai:jager.gerold|jager.gerhard.1|jager.gabriela|jager.gerhard|jager.gunther|jager.georg(no abstract)Structures induced by Alexandrov fuzzy topologieshttps://zbmath.org/1522.540132023-12-07T16:00:11.105023Z"Kim, Yong Chan"https://zbmath.org/authors/?q=ai:kim.yong-chan.1|kim.yongchan|kim.yong-chan(no abstract)The topological structure of the set of fuzzy numbers with the supremum metrichttps://zbmath.org/1522.540142023-12-07T16:00:11.105023Z"Liu, Dongming"https://zbmath.org/authors/?q=ai:liu.dongming"Yang, Zhongqiang"https://zbmath.org/authors/?q=ai:yang.zhongqiang"Zhao, Dongsheng"https://zbmath.org/authors/?q=ai:zhao.dongshengSummary: We study the family of all fuzzy sets of the \(n\)-dimensional Euclidean space, which are upper-semicontinuous, fuzzy convex and normal with compact supports contained in a non-degenerate convex subset Y, and prove that the following statements are equivalent: (i) The family of fuzzy sets with the topology induced by the supremum metric is homeomorphic to a non-separable Hilbert space whose weight is the cardinality of the set of all real numbers; (ii) the non-degenerate convex subset Y is topologically complete, or equivalently, it is a countable intersection of open sets in the n-dimensional Euclidean space.The topological structure of the space of fuzzy compactahttps://zbmath.org/1522.540152023-12-07T16:00:11.105023Z"Liu, Wenjuan"https://zbmath.org/authors/?q=ai:liu.wenjuan"Yang, Hanbiao"https://zbmath.org/authors/?q=ai:yang.hanbiao"Yang, Zhongqiang"https://zbmath.org/authors/?q=ai:yang.zhongqiangSummary: A \textbf{fuzzy compactum} in a space \(X\) is an upper semi-continuous map \(f:A\to [0,1]\) from a nonempty compact subspace \(A \subset X\) to the unit closed interval \([0,1]\) such that \(\{x \in A : f(x) > 0\}\) is dense in \(A\). In this paper, we investigate the topological structure of the space \(\mathcal{K}_e(X)\) of fuzzy compacta of a separable metric space \(X\) with the Hausdorff metric. In particular, we prove that \(\mathcal{K}_e(X) \approx \ell^2\) if \(X\) is an infinite, locally compact, locally connected, connected and separable metric space.A note regarding some fuzzy SSPO mappingshttps://zbmath.org/1522.540162023-12-07T16:00:11.105023Z"Makolli, Shkumbin"https://zbmath.org/authors/?q=ai:makolli.shkumbin"Krsteska, Biljana"https://zbmath.org/authors/?q=ai:krsteska.biljanaSummary: In this paper we will introduce the concept of fuzzy SSPO irresolute continuous mappings, fuzzy SSPO irresolute open (closed) mappings as
well as the concept of SSPO homeomorphism. We will also study some of their properties and their relation with other forms of fuzzy continuity.Betweenness relations and gated sets in fuzzy metric spaceshttps://zbmath.org/1522.540172023-12-07T16:00:11.105023Z"Shi, Yi"https://zbmath.org/authors/?q=ai:shi.yi.1Summary: Gated sets in metric spaces play an important role in the study of convexity. In this paper, we investigate the notion of gated sets in the context of betweenness of fuzzy metric spaces. For this purpose, we analyze the validity of postulates presented by \textit{E. V. Huntington} and \textit{J. R. Kline} [Trans. Am. Math. Soc. 18, 301--325 (1917; JFM 46.1429.02)] to the betweenness in GV fuzzy metric spaces. The postulates are different from the usual properties of metric betweenness, amounting to thirteen. Interestingly, the validity of most of these postulates to the betweenness need to be considered in fuzzy normed spaces. In particular, two of the postulates for this relation are valid when the space is 1-dimensional. Five of the postulates for this relation are centrally studied in connection with strict convexity. As a consequence, several characterizations on strict convexity in a GV fuzzy normed space under the minimum t-norm are established.Characterizations of \(L\)-order \(L\)-convex spaceshttps://zbmath.org/1522.540182023-12-07T16:00:11.105023Z"Su, S. H."https://zbmath.org/authors/?q=ai:su.shuhua"Li, Q. G."https://zbmath.org/authors/?q=ai:li.qingguo"Liu, F. Y."https://zbmath.org/authors/?q=ai:liu.fuyao"Li, Q."https://zbmath.org/authors/?q=ai:li.qi(no abstract)T-complete KM-fuzzy metric spaces via domain theoryhttps://zbmath.org/1522.540192023-12-07T16:00:11.105023Z"Wang, Kai"https://zbmath.org/authors/?q=ai:wang.kai.13|wang.kai.2|wang.kai.3|wang.kai.5|wang.kai.9|wang.kai.1|wang.kai.4|wang.kai.6|wang.kai.8Summary: In this paper we give a new method of constructing partial order on the set of formal balls of a KM-fuzzy metric space, which overcomes a key defect that the partial order on the set of formal balls depends on the minimum t-norm \(\wedge\) in the case of
[\textit{I. Mardones-Pérez} and \textit{M. A. de Prada Vicente}, ibid. 300, 72--83 (2016; Zbl 1378.54010); \textit{L. A. Ricarte} and \textit{S. Romaguera}, Topology Appl. 163, 149--159 (2014; Zbl 1312.54003)]. Further, based on a continuous t-norm, we will establish a connection between KM-fuzzy metric space and domain theory, which is that a KM-fuzzy metric space is T-complete if and only if its poset of formal balls is a domain.\((L,M)\)-fuzzy topological derived internal relations and \((L,M)\)-fuzzy topological derived enclosed relationshttps://zbmath.org/1522.540202023-12-07T16:00:11.105023Z"Wu, X. Y."https://zbmath.org/authors/?q=ai:wu.xingying|wu.xiangyi|wu.xian-yuan|wu.xingyu|wu.xuyang|wu.xinyu|wu.xiongying|wu.xianyin|wu.xinyou|wu.xianyan|wu.xiao-yuan|wu.xinying|wu.xiangyao|wu.xianyu|wu.xingyao|wu.xueyan|wu.xiaoying|wu.xianyue|wu.xinyue|wu.xinyi|wu.xiaoyang|wu.xueyuan|wu.xiuyun|wu.xinye|wu.xiangyang|wu.xiaoyong|wu.xue-yun|wu.xiangyu|wu.xiaoyue|wu.xueying|wu.xinyun|wu.xinyang|wu.xuanyu|wu.xiyuan|wu.xianyong|wu.xinyuan|wu.xingyuan|wu.xianyi|wu.xiaoyun|wu.xingye|wu.xiangyun|wu.xiaoyan|wu.xuyi|wu.xiaoyu"Shi, Y."https://zbmath.org/authors/?q=ai:shi.yi(no abstract)A categorical isomorphism between injective balanced \(L\)-\(S_0\)-convex spaces and fuzzy frameshttps://zbmath.org/1522.540212023-12-07T16:00:11.105023Z"Xia, Changchun"https://zbmath.org/authors/?q=ai:xia.changchunSummary: The main purpose of this paper is to show that injective balanced \(L\)-\(S_0\)-convex spaces and fuzzy frames are isomorphic from the categorical point of view. Meanwhile, we get that every fuzzy frame equipped with the strong \(L\)-filter convex structure is an injective balanced \(L\)-\(S_0\)-convex space and conversely, the specialization \(L\)-ordered set of an injective balanced \(L\)-\(S_0\)-convex space is a fuzzy frame.Fuzzy betweenness spaces on continuous latticeshttps://zbmath.org/1522.540222023-12-07T16:00:11.105023Z"Zhang, S. Y."https://zbmath.org/authors/?q=ai:zhang.shuyou|zhang.shuyun|zhang.shenyuan|zhang.shiyun|zhang.shangyuan|zhang.sara-ying|zhang.shai-yan|zhang.shaoyong|zhang.suyan|zhang.siyao|zhang.shuyu.1|zhang.shiying|zhang.shuangyin|zhang.siyan|zhang.siying|zhang.shu-yan|zhang.shuoying|zhang.shouyuan|zhang.shangyou|zhang.shengyu.1|zhang.shuyong|zhang.shuying|zhang.sheng-yun|zhang.songyan|zhang.suyong|zhang.suying|zhang.siyun|zhang.shengyuan.1|zhang.shenyu|zhang.saiyin|zhang.shaoyi|zhang.shiyu|zhang.shuiying|zhang.sanyuan|zhang.saiyan|zhang.shuyuan|zhang.shengyuan|zhang.shaoyi.1|zhang.shanyuan|zhang.siyu|zhang.shuyi|zhang.shuangyue|zhang.shiyong|zhang.shenyong|zhang.shunyan|zhang.siyuan|zhang.siyi|zhang.suyang|zhang.shuaiyin|zhang.shaoyu|zhang.shanying|zhang.shaoyang|zhang.shangyao|zhang.senyue|zhang.shaoyun|zhang.suyu|zhang.shiyang|zhang.siyang|zhang.shaoyan"Shi, F. G."https://zbmath.org/authors/?q=ai:shi.fu-gui(no abstract)On fuzzy monotone convergence \(\mathcal{Q}\)-cotopological spaceshttps://zbmath.org/1522.540232023-12-07T16:00:11.105023Z"Zhang, Zhongxi"https://zbmath.org/authors/?q=ai:zhang.zhongxi"Shi, Fu-Gui"https://zbmath.org/authors/?q=ai:shi.fu-gui"Li, Qingguo"https://zbmath.org/authors/?q=ai:li.qingguo"Wang, Kai"https://zbmath.org/authors/?q=ai:wang.kai.13Summary: In this paper, we generalize the concept of a monotone convergence space (also called a \(d\)-space) to the setting of a \(\mathcal{Q}\)-cotopological space, where \(\mathcal{Q}\) is a commutative and integral quantale. We establish a \(D\)-completion for every stratified \(\mathcal{Q} \)-cotopological space, which is a category reflection of the category \(\mathbf{S} \mathcal{Q}\)-\(\mathbf{CTop}\) of stratified \(\mathcal{Q}\)-cotopological spaces onto the full subcategory \(\mathbf{S}\mathcal{Q}\)-\(\mathbf{DCTop}\) of monotone convergence \(\mathcal{Q}\)-cotopological spaces. By introducing the notion of a tapered set, a direct characterization of the completion is obtained: the \(D\)-completion of each stratified \(\mathcal{Q}\)-cotopological space \(X\) consists exactly of those tapered closed sets in \(X\). We show that the \(D\)-completion can be applied to obtain a universal fuzzy directed completion of a \(\mathcal{Q}\)-ordered set by endowing it with the Scott cotopology, taking the \(D\)-completion, and then passing to the specialization \(\mathcal{Q}\)-order. Consequently, the category \(\mathcal{Q}\)-\(\mathbf{DOrd}\) of fuzzy directed complete \(\mathcal{Q}\)-ordered sets and Scott continuous functions is reflective in the category \(\mathcal{Q}\)-\(\mathbf{Ord}_\sigma\) of \(\mathcal{Q}\)-ordered sets and Scott continuous functions.When is the space of semi-additive functionals an absolute (neighbourhood) retract?https://zbmath.org/1522.540242023-12-07T16:00:11.105023Z"Zaitov, Adilbek"https://zbmath.org/authors/?q=ai:zaitov.adilbek-atakhanovich"Kurbanov, Khamidjon"https://zbmath.org/authors/?q=ai:kurbanov.khamidjonSummary: In the present paper it is proved that if for a given compact Hausdorff space \(X\) the hyperspace \(exp(X)\) is a contractible compact space then the space \(OS_f(X)\) is also a contractible compact space. As a consequence it is established that the space \(OS_f(X)\) of semi-additive functionals is an absolute (neighbourhood) retract if and only if the hyperspace \(exp(X)\) is so.More on decomposition of bioperation-continuityhttps://zbmath.org/1522.540252023-12-07T16:00:11.105023Z"Carpintero, Carlos"https://zbmath.org/authors/?q=ai:carpintero.carlos-r"Rajesh, Neelamegarajan"https://zbmath.org/authors/?q=ai:rajesh.neelamegarajan"Rosas, Ennis"https://zbmath.org/authors/?q=ai:rosas.ennis-rSummary: In this paper, we define the notions of weakly \((\gamma, \gamma^\prime)\)-\((\beta, \beta^\prime)\)-continuity and weakly\(^\ast\) \((\gamma, \gamma^\prime)\)-\((\beta, \beta^\prime)\)-continuity in bioperation-topological spaces and investigate some of their properties and relationships with \((\gamma, \gamma^\prime)\)-\((\beta, \beta^\prime)\)-continuity.Image of quasicomponents by proximately chain refinable functionshttps://zbmath.org/1522.540262023-12-07T16:00:11.105023Z"Buklla, Abdulla"https://zbmath.org/authors/?q=ai:buklla.abdullaSummary: Proximate chain refinable functions were defined in [the author and \textit{G. Markoski}, Hacet. J. Math. Stat. 48, No. 5, 1437--1442 (2019; Zbl 1488.54056)] as a coarser class of functions from proximate refinable maps. In this paper we investigate some properties of these functions, we show that proximately chain
refinable functions map quasicomponents to quasicomponents if codomain is with open connected components and in the end we provide some counterexamples.On condensations onto \(\sigma \)-compact spaceshttps://zbmath.org/1522.540272023-12-07T16:00:11.105023Z"Lipin, A. E."https://zbmath.org/authors/?q=ai:lipin.anton-evgenevich"Osipov, A. V."https://zbmath.org/authors/?q=ai:osipov.a-v|osipov.alexandr-v|osipov.andrei-v|osipov.alexander-v|osipov.aleksey-v|osipov.andrey-vThe aim of the paper under review is to prove the following result. Let \(X\) be a complete metric space of weight \(w(X)\) and \(H\subseteq X\) be a subset such that \(w(X)<|H|<\mathfrak{c}\). Then there is no continuous bijection (condensation) of the subspace \(X\setminus H\) onto a \(\sigma\)-compact space. The argument of the proof is contained in Section 3 and 4, where several auxiliary and technical results are proved. In Section 5 some consequences of the main result are presented. Among them we highlight Corollary 2 that states as follows: Let \(X\) be a complete metric space and \(H\subseteq X\) be such that \(w(X)<|H|<\mathfrak{c}\). Then the subspace \(X\setminus H\) does not condense onto a separable absolute Borel space (a space homeomorphic to a Borel subset of a complete metric space). From this result the following problem raises naturally (Question 2):
Does there exist a complete metric space \(X\) and a set \(H\subseteq X\) such that \(w(X)<|H|<\mathfrak{c}\) and the subspace \(X\setminus H\) condenses onto a complete metric space?
Reviewer: Jacopo Somaglia (Milano)On the existence of isovariant extensors for actions of locally compact groupshttps://zbmath.org/1522.540282023-12-07T16:00:11.105023Z"Ageev, Sergei"https://zbmath.org/authors/?q=ai:ageev.sergei-m"Bykov, Alexander"https://zbmath.org/authors/?q=ai:bykov.alexander-i|bykov.aleksandr-nikolaevichIn their Introduction, the authors give some background about this particular area of study. They speak in particular of Palais' problem and its solution for the class of \(G\)-spaces that are metrizable when \(G\) is a compact group. The new result, which we will state later, involves locally compact Hausdorff groups.
In Section 1, the reader will find a concise, nicely written summary of the concept of a \(G\)-space and the most basic notions that one would need to know in order to comprehend the ensuing theoretical developments. In particular the definition of a \textit{proper} \(G\)-space is given along with some fundamental facts about proper \(G\)-spaces. Also the notion of isovariance is explained. The class \(G\)-\(\mathcal{PM}\) is defined. These are proper \(G\)-spaces \(\mathbb{X}\) such that the orbit space \(\mathbb{X}/G\) is metrizable.
Sections 2-5 involve building up the theory that will be needed later. In Section 6, a certain space \(\mathbf{H}\), which depends on a given locally compact Hausdorff group \(G\), is described; it contains a specified point \(*\). From this, one defines the important space \(\mathbf{H}^*=\mathbf{H}\setminus\{*\}\) that will appear in the main result:
\textbf{Theorem 7.1.} Let \(G\) be a locally compact Hausdorff group. Then \(\mathbf{H}^*\) is a \(G\)-\(\mathcal{PM}\)-space and an isovariant absolute extensor for the class \(G\)-\(\mathcal{PM}\).
Reviewer: Leonard R. Rubin (Norman)On the \(l_p\)-equivalence of ultrafiltershttps://zbmath.org/1522.540292023-12-07T16:00:11.105023Z"Baars, Jan"https://zbmath.org/authors/?q=ai:baars.janThe set-theoretic framework for this work is ZFC. All spaces are assumed to be Tikhonov. For a space \(X\), \(C(X)\) stands for the set of all real-valued continuous functions on \(X\). If \(C(X)\) is considered as a topological subspace of \(\mathbb{R}^{X}\), it is denoted by \(C_p(X)\). That is, \(C_p(X)\) stands for \(C(X)\) equipped with the topology of pointwise convergence. The Tikhonov spaces \(X\) and \(Y\) are called \(l_p\)-equivalent if the topological vector spaces \(C_p(X)\) and \(C_p(Y)\) are linearly homeomorphic. For the denumerable discrete space \(\omega\) of all finite ordinals, \(\omega^{\ast}\) is the remainder of the Čech-Stone compactification \(\beta\omega\) of \(\omega\). For every \(u\in\omega^{\ast}\), \(\omega_u\) denotes the subspace \(\omega\cup\{u\}\) of \(\beta\omega\). The author shows a gap in the following factorization theorem by Gul'ko on \(C_p(\omega_u)\) (Theorem 3 in [\textit{S. P. Gul'ko}, Math. Notes 47, No. 4, 329--334 (1990; Zbl 0719.54017)]): If \(C_p(\omega_u)\) is a direct sum of its two infinitely dimensional closed vector subspaces, then one of the subspaces is linearly homeomorphic to \(C_p(\omega_u)\) and the other is linearly homeomorphic to \(\mathbb{R}^{\aleph_0}\).
The author gives generalizations and, not depending on the factorization theorem, new proofs of some results from the above-mentioned paper by Gul'ko. In particular, by applying the Rudin-Keisler (pre-)order on \(\beta\omega\), Hasumi's result from [\textit{M. Hasumi}, Math. Ann. 179, 83--89 (1969; Zbl 0167.51103)] and partly Gul'ko's methods, an alternative proof of the following theorem by Gul'ko is presented: For all \(u, v\in\omega^{\ast}\), it holds that the spaces \(\omega_u\) and \(\omega_v\) are \(l_p\)-equivalent if and only if they are homeomorphic. The following generalization of this theorem is obtained: For positive integers \(n,m\) and a subset \(\{u_1,\dots, u_n, v_1,\dots, v_m\}\) of \(\omega^{\ast}\), the spaces \(\bigoplus_{i=1}^{n}\omega_{u_i}\) and \(\bigoplus_{i=1}^{m}\omega_{v_i}\) are \(l_p\)-equivalent if and only if \(n=m\) and there is a permutation \(\pi\) of the set \(\{1,\dots, n\}\) such that, for every \(i\in\{1,\dots n\}\), the spaces \(\omega_{u_i}\) and \(\omega_{v_{\pi(i)}}\) are homeomorphic. This implies that, for every pair \(m,n\) of distinct positive integers and for every \(u\in\omega^{\ast}\), the spaces \(C_p(\omega_u)^{n}\) and \(C_p(\omega_u)^{m}\) are not linearly homeomorphic. The author remarks that the latter result was also stated in the above-mentioned paper by Gul'ko but as a corollary to Gul'ko's factorization theorem which seems to have an incorrect proof. Furthermore, the author proves that, for every pair \(n,m\) of positive integers, a subset \(\{u_1,\dots u_n\}\) of \(\omega^{\ast}\) and countable Tikhonov spaces \(Y_1,\dots Y_n\) such that, for every \(i\in\{1,\dots m\}\), \(Y_i\) has only one non-isolated point, and the spaces \(\bigoplus_{i=1}^{n}\omega_{u_i}\) and \(\bigoplus_{i=1}^{m}Y_i\) are \(l_p\)-equivalent, the inequality \(m\leq n\) holds; moreover, \(n=m\) if and only if, for every \(i\in\{1,\dots n\}\), there exists \(v_i\in\omega^{\ast}\) such that \(Y_i\) is homeomorphic to \(\omega_{v_i}\). This implies that, for every countable Tikhonov space \(Y\) with only one non-isolated point and every \(u\in\omega^{\ast}\), the spaces \(Y\) and \(\omega_u\) are \(l_p\)-equivalent if and only if they are homeomorphic; thus, if \(Y\) and \(\omega_u\) are \(l_p\)-equivalent and \(y_0\) is the unique non-isolated point of \(Y\), then the family \(\mathcal{U}=\{U\subseteq Y\setminus\{y_0\}: y_0\in\text{cl}_{Y}U\}\) is a free ultrafilter. This gives an alternative, detailed proof of another claim by Gul'ko.
In Question 5.1, the author asks if Gul'ko's factorization theorem is true. This question is left as an open problem posed as follows: ``Let \(u\in\omega^{\ast}\) and let \(C_p(\omega_u)=E\times F\), where \(E\) and \(F\) are closed linear subspaces of \(C_p(\omega_u)\). Is it true that either \(E\) or \(F\) is linearly homeomorphic to \(C_p(\omega_u)\) and the other to \(\mathbb{R}^{\infty}\)?''
Finally, it is observed that, for every pair \(n,m\) of positive integers with \(n\neq m\), there exist countable Tikhonov spaces \(X_1,\dots ,X_n, Y_1,\dots ,Y_m\) with only one non-isolated point each, such that the spaces \(\bigoplus_{i=1}^{n}X_i\) and \(\bigoplus_{i=1}^m Y_i\) are \(l_p\)-equivalent. A deeper analysis of this observation is given. Assuming that \(X,Y\) are \(l_p\)-equivalent countable Tikhonov spaces with only one non-isolated point each, the author asks if \(X\) must be homeomorphic to \(Y\) (Question 5.3).
\textit{Reviewer's remark:} In Question 5.1, it should be assumed that the spaces \(E\) and \(F\) are both infinite dimensional.
Reviewer: Eliza Wajch (Siedlce)On subspaces of spaces \(C_p(X)\) isomorphic to spaces \(c_0\) and \(\ell_q\) with the topology induced from \(\mathbb{R}^{\mathbb{N}}\)https://zbmath.org/1522.540302023-12-07T16:00:11.105023Z"Kąkol, Jerzy"https://zbmath.org/authors/?q=ai:kakol.jerzy"Molto, Anibal"https://zbmath.org/authors/?q=ai:molto.anibal"Śliwa, Wiesław"https://zbmath.org/authors/?q=ai:sliwa.wieslawThe authors study spaces of continuous, real-valued functions on completely regular (Tychonoff) spaces endowed with the topology of pointwise convergence and the question when they contain copies of the space \(c_0\), again with the topology of pointwise convergence (denoted \((c_0)_p\)). They prove that for an infinite completely regular space \(X\), \(C_p(X)\) does indeed contain a copy of \((c_0)_p\). Moreover, if \(X\) contains an infinite compact subset, then \(C_p(X)\) contains a closed copy of \((c_0)_p\).
Reviewer: Tomasz Kania (Praha)Spaces of functions of the first Baire class in the topology of pointwise convergence and their \(l\)-equivalencehttps://zbmath.org/1522.540312023-12-07T16:00:11.105023Z"Khmyleva, T. E."https://zbmath.org/authors/?q=ai:khmyleva.tatana-e"Genze, L. V."https://zbmath.org/authors/?q=ai:genze.leonid-v(no abstract)On polynomial homeomorphisms of spaces of continuous functionshttps://zbmath.org/1522.540322023-12-07T16:00:11.105023Z"Lazarev, V. R."https://zbmath.org/authors/?q=ai:lazarev.vadim-remirovich(no abstract)On some equivalences on the class of Tikhonov spaceshttps://zbmath.org/1522.540332023-12-07T16:00:11.105023Z"Lazarev, V. R."https://zbmath.org/authors/?q=ai:lazarev.vadim-remirovich(no abstract)Connectedness modulo an ideal; a characterization theoremhttps://zbmath.org/1522.540342023-12-07T16:00:11.105023Z"Koushesh, M. R."https://zbmath.org/authors/?q=ai:koushesh.m-rThis research is conducted in ZFC. Assume that \(X\) is a non-empty Tychonoff space and \(\mathscr{I}\) is an ideal of subsets of \(X\). A function \(f: X\to [0,1]\) is called 2-valued modulo \(\mathscr{I}\) if \(f^{-1}(0)\notin\mathscr{I}\), \(f^{-1}(1)\notin\mathscr{I}\) but \(X\setminus(f^{-1}(0)\cup f^{-1}(1))\in\mathscr{I}\). The space \(X\) is called connected modulo \(\mathscr{I}\) if there does not exist a 2-valued modulo \(\mathscr{I}\) continuous function \(f:X\to [0, 1]\). Let \(C(X)\) be the ring of all continuous real functions defined on \(X\). The ring of all bounded functions from \(C(X)\) is denoted by \(C_B(X)\). The set \(C_0^{\mathscr{I}}(X)=\{f\in C_B(X): (\forall \epsilon >0) \vert f\vert^{-1}([\epsilon, +\infty))\in\mathscr{I}\}\) is an ideal of the ring \(C_B(X)\). A ring \(R\) is called indecomposable if there does not exist a pair \(I_1, I_2\) of proper ideals of \(R\) such that \(R=I_1\oplus I_2\).
The main theorem of this article states that the space \(X\) is connected modulo \(\mathscr{I}\) if and only if the quotient ring \(C_B(X)/C_0^{\mathscr{I}}(X)\) is indecomposable. The proof that the indecomposability of \(C_B(X)/C_0^{\mathscr{I}}(X)\) implies that \(X\) is connected modulo \(\mathscr{I}\) is algebraic. To prove that if \(X\) is connected modulo \(\mathscr{I}\), then the ring \(C_B(X)/C_0^{\mathscr{I}}(X)\) is indecomposable, the author uses the Čech-Stone compactification \(\beta X\) of \(X\), its subspace \(\lambda_{\mathscr{I}}X=\bigcup\{\text{int}_{\beta X}\text{cl}_{\beta X} A: A\subseteq X\wedge \text{cl}_X A\in\mathscr{I}\}\) and the following theorem proved in [\textit{M. R. Koushesh}, Topology Appl. 214, 150--179 (2016; Zbl 1371.54137)]: \(X\) is connected modulo \(\mathscr{I}\) if and only if the space \(\beta X\setminus \lambda_{\mathscr{I}}X\) is connected. The above-mentioned conditions equivalent to the connectivity modulo \(\mathscr{I}\) are applied to various choices of ideals of subsets of the space \(X\).
For instance:
\begin{itemize}
\item[(1)] if \(\mathscr{N}=\{\emptyset\}\), then \(C_B(X)\) is isomorphic to the quotient ring \(C_B(X)/C_0^{\mathscr{N}}(X)\), so \(X\) is connected if and only if \(C_B(X)\) is indecomposable (if and only if \(\beta X\) is connected);
\item[(2)] if \(\mathscr{K}\) is the family of subsets of \(X\) having compact closures in \(X\), then \(C_0^{\mathscr{K}}(X)=C_0(X)\), \(X\) is connected modulo \(\mathcal{K}\) if and only if \(C_B(X)/C_0(X)\) is indecomposable; in consequence, if \(X\) is locally compact, then \(C_B(X)/C_0(X)\) is indecomposable if and only if \(\beta X\setminus X\) is connected;
\item[(3)] if \(\mathscr{B}=\{A\subseteq X: (\forall f\in C(X)) f(A)\text{ is bounded }\}\) and \(vX\) is the Hewitt realcompactification of \(X\), then \(X\) is connected modulo \(\mathscr{B}\) if and only if \(\text{cl}_{\beta X}(\beta X\setminus vX)\) is connected (if and only if \(C_B(X)/C_0^{\mathscr{B}}(X)\) is indecomposable);
\item[(4)] if \(\mathscr{S}\) if the family of all sets \(A\subseteq X\) such that \(A\) has a pseudocompact neighborhood in \(X\), then \(\mathscr{S}\) is an ideal of subsetes of \(X\), \(\lambda_{\mathscr{S}}X=\text{int}_{\beta X}vX\), \(C_B(X)/C_0^{\mathscr{S}}(X)\) is indecomposable if and only if \(\text{cl}_{\beta X}(\beta X\setminus vX)\) is connected;
\item[(5)] if \(X\) is a normal space, \(\mathscr{R}\) is the ideal of all subsets of \(X\) whose closures in \(X\) are realcompact, then \(C_B(X)/C_0^{\mathscr{R}}(X)\) is indecomposable if and only if \(\text{cl}_{\beta X}(vX\setminus X)\) is connected.
\end{itemize}
Given a Borel measure \(\mu\) on \(X\), the ideal of Borel sets \(A\) in \(X\) such that \(\mu(A)=0\) is also discussed. Finally, it is observed that if \(X\) is normal, then, for \(\mathscr{I}=\mathscr{B}\) or \(\mathscr{I}=\mathscr{S}\), the equality \(C_0^{\mathscr{I}}(X)\cap C_0^{\mathscr{R}}(X)=C_0(X)\) holds and, in addition, if \(X\) is also locally compact, then \(C_0^{\mathscr{I}}(X)C_0^{\mathscr{R}}(X)=C_0(X)\).
Reviewer: Eliza Wajch (Siedlce)Some observations on a clopen version of the Rothberger propertyhttps://zbmath.org/1522.540352023-12-07T16:00:11.105023Z"Bhardwaj, Manoj"https://zbmath.org/authors/?q=ai:bhardwaj.manoj"Osipov, Alexander V."https://zbmath.org/authors/?q=ai:osipov.alexander-v|osipov.alexandr-vLet \(\mathcal{O}\) be the collection of all open covers of \(X\) and \(\mathcal{C}_{\mathcal{O}}\) be the collection of all clopen covers of \(X\). In this paper, the authors prove that a clopen version \(S_{1}(\mathcal{C}_{\mathcal{O}}, \mathcal{C}_{\mathcal{O}})\) of the Rothberger property and Borel strong measure zeroness are independent. For a zero-dimensional metric space \((X, d)\), \(X\) satisfies \(S_{1}(\mathcal{C}_{\mathcal{O}}, \mathcal{C}_{\mathcal{O}})\) if and only if \(X\) has Borel strong measure zero with respect to each metric which has the same topology as \(d\) has. In a zero-dimensional space, the game \(G_{1}(\mathcal{O}, \mathcal{O})\) is equivalent to the game \(G_{1}(\mathcal{C}_{\mathcal{O}}, \mathcal{C}_{\mathcal{O}})\) and the point-open game is equivalent to the point-clopen game. Using reflections, the authors obtain that the game \(G_{1}(\mathcal{C}_{\mathcal{O}}, \mathcal{C}_{\mathcal{O}})\) and the point-clopen game are strategically and Markov dual. An example is given for a space on which the game \(G_{1}(\mathcal{C}_{\mathcal{O}}, \mathcal{C}_{\mathcal{O}})\) is undetermined.
The authors prove the following results.
\textbf{Theorem 3.1.} For a zero-dimensional space \(X\), \(S_{1}(\mathcal{C}_{\mathcal{O}}, \mathcal{C}_{\mathcal{O}})\) is equivalent to \(S_{1}(\mathcal{O}, \mathcal{O})\).
\textbf{Theorem 3.13.} For a zero-dimensional space, the following statements hold:
(1) The game \(G_{1}(\mathcal{C}_{\mathcal{O}}, \mathcal{C}_{\mathcal{O}})\) is equivalent to the game \(G_{1}(\mathcal{O}, \mathcal{O})\).
(2) The point-clopen game is equivalent to the point-open game.
Reviewer: Zuquan Li (Hangzhou)Further investigations on certain star selection principleshttps://zbmath.org/1522.540362023-12-07T16:00:11.105023Z"Chandra, Debraj"https://zbmath.org/authors/?q=ai:chandra.debraj"Alam, Nur"https://zbmath.org/authors/?q=ai:alam.nurThe study of star selection principles is well known and recently several papers have been devoted to this topic. In this article the authors consider certain star versions of the Menger, Hurewicz and Rothberger properties. A variety of investigations is performed using Alster covers and critical cardinalities (\(\mathfrak{d}\), \(\mathfrak{b}\) and \(\mathrm{cov}({\mathcal M})\)). Interesting observations on the extent and the Alexandroff duplicate are given.
Reviewer: Maddalena Bonanzinga (Messina)On monotonically star countably compact spaceshttps://zbmath.org/1522.540372023-12-07T16:00:11.105023Z"Singh, Sumit"https://zbmath.org/authors/?q=ai:singh.sumit|singh.sumit.1If \(\mathcal U\) is a cover of a topological space \(X\) and \(A\subset X\), then \(\text{St}(A,X)= \bigcup\{U: U\in \mathcal U\) and \(U\cap A\neq \emptyset\}\). A cover \(\mathcal V\) of the space \(X\) refines \(\mathcal U\) if, for every \(V\in \mathcal V\), there exists \(U\in \mathcal U\) such that \(V\subset U\). A space \(X\) is called monotonically star countably compact if it is possible to assign to every open cover \(\mathcal U\) of the space \(X\) a countably compact subspace \(s(\mathcal U) \subset X\) in such a way that \(\text{St}(s(\mathcal U), \mathcal U)= X\) and if \(\mathcal V\) refines \(\mathcal U\), then \(s(\mathcal U) \subset s(\mathcal V)\).
It is established in the paper that monotonical star countable compactness is preserved by continuous images and every Tychonoff space embeds as a closed subspace in a monotonically star countably compact space. An example is given of a monotonically star countably compact space with a regular closed subspace which is not monotonically star countaby compact. It is also shown that a Tychonoff monotonically star countably compact space need not be monotonically star Lindelöf and the product of two countably compact spaces can fail to be monotonically star countably compact.
Reviewer: Vladimir Tkachuk (Ciudad de México)A note on the Freese-Nation property and topological gameshttps://zbmath.org/1522.540382023-12-07T16:00:11.105023Z"Bąk, Judyta"https://zbmath.org/authors/?q=ai:bak.judytaSummary: In this paper we present an alternative proof, using some topological game, of the fact that compact spaces with the separative Freese-Nation property for a base consisting of co-zero sets are openly generated; see [\textit{J. Bąk} and \textit{A. Kucharski}, Topology Appl. 304, Article ID 107784, 11 p. (2021; Zbl 1479.54052)].
For the entire collection see [Zbl 1518.26001].Some properties of ultrafilters related to their use as generalized elementshttps://zbmath.org/1522.540392023-12-07T16:00:11.105023Z"Chentsov, A. G."https://zbmath.org/authors/?q=ai:chentsov.alexander-gSummary: Ultrafilters of broadly understood measurable spaces and their application as generalized elements in abstract reachability problems with constraints of asymptotic nature are considered. Constructions for the immersion of conventional solutions, which are points of a fixed set, into the space of ultrafilters and representations of ``limit'' ultrafilters realized with topologies of Wallman and Stone types are studied. The structure of the attraction set is established using constraints of asymptotic nature in the form of a nonempty family of sets in the space of ordinary solutions. The questions of implementation up to any preselected neighborhood of the attraction sets in the topologies of Wallman and Stone types are studied. Some analogs of the mentioned properties are considered for the space of maximal linked systems.Metric segments in Gromov-Hausdorff classhttps://zbmath.org/1522.540402023-12-07T16:00:11.105023Z"Borisova, Ol'ga Borisovna"https://zbmath.org/authors/?q=ai:borisova.olga-borisovnaSummary: We study properties of metric segments in the class of all metric spaces considered up to an isometry, endowed with Gromov-Hausdorff distance. On the isometry classes of all compact metric spaces, the Gromov-Hausdorff distance is a metric. A metric segment is a class that consists of points lying between two given ones. By von Neumann-Bernays-Gödel (NBG) axiomatic set theory, a proper class is a ``monster collection'', e.g., the collection of all sets. We prove that any metric segment in the proper class of isometry classes of all metric spaces with the Gromov-Hausdorff distance is a proper class if the segment contains at least one metric space at positive distances from the segment endpoints. If the distance between the segment endpoints is zero, then the metric segment is a set. In addition, we show that the restriction of a non-degenerated metric segment to compact metric spaces is a non-compact set.Compact mappings and \(s\)-mappings at subsetshttps://zbmath.org/1522.540412023-12-07T16:00:11.105023Z"Lin, Shou"https://zbmath.org/authors/?q=ai:lin.shou"Ling, Xuewei"https://zbmath.org/authors/?q=ai:ling.xuewei"He, Wei"https://zbmath.org/authors/?q=ai:he.wei.2A subset \(A\) of a topological space \((X, \tau)\) is said to be a sequential neighborhood of a point \(x \in X\) if each sequence convergent to \(x\) is eventually in \(A\). If \(A\) is a sequential neighborhood for every \(a \in A\), then \(A\) is sequentially open. The space \((X, \tau)\) is called a sequential space if \(S \in \tau\) holds for every sequentially open \(S \subseteq X\).
\textbf{Definition.} Let \(\mathcal{P}\) be a family of subsets of a topological space \((X, \tau)\) and let \(A \subseteq X\). The family \(\mathcal{P}\) is a \(cs^*\)-network at \(A\) if, for each \(x \in A\), every sequence \((x_n)_{n \in \mathbb{N}}\) converging to \(x\) in \((X, \tau)\), and every open neighborhood \(U \in \tau\) of the point \(x\), there exist a subsequence \((x_{n_j})_{j \in \mathbb{N}}\) of \((x_n)_{n \in \mathbb{N}}\) and \(P \in \mathcal{P}\) such that
\[
\{x\} \cup \{x_{n_j} \colon j \in \mathbb{N}\} \subset P \subset U.
\]
A \(cs^*\)-network \(\mathcal{P}\) at \(A\) is point-regular at \(A\) if, for each \(x \in A\) and every open neighborhood \(U \in \tau\) of \(x\), the family
\[
\{P \in \mathcal{P} \colon P \not\subset U \text{ and } P \ni x\}
\]
is finite.
The following theorem of the authors gives an affirmative answer to Question~4.9 from [\textit{S. Lin} et al., Topology Appl. 271, Article ID 106987, 13 p. (2020; Zbl 1437.54012)].
\textbf{Theorem.} The following statements are equivalent for every topological space \((X, \tau)\).
\begin{itemize}
\item[1.] \((X, \tau)\) is a sequential space with a point-regular \(cs^*\)-network at the set of all non-isolated points of \((X, \tau)\).
\item[2.] \textit{There are a metric space \(M\) and a quotient mapping \(f \colon M \to X\) such that \(f^{-1}(x)\) is a compact subset of \(M\) for every non-isolated point \(x \in X\)}.
\end{itemize}
In Example~4.1 of the paper it is shown that the Aleksandroff double line space \(X\) has no base which is point countable at the set of all non-isolated points of \(X\), but has a point-regular \(cs^*\)-network at this set. In particular, Example~4.1 gives a negative answer to Question~3.7 from [\textit{X. Ling} et al., ibid. 290, Article ID 107572, 18 p. (2021; Zbl 1465.54023)].
The article also contains other non-trivial results related to quotient images of metric spaces.
Reviewer: Aleksey A. Dovgoshey (Slovyansk)Completeness problem via fixed point theoryhttps://zbmath.org/1522.540422023-12-07T16:00:11.105023Z"Anjum, Rizwan"https://zbmath.org/authors/?q=ai:anjum.rizwan"Abbas, Mujahid"https://zbmath.org/authors/?q=ai:abbas.mujahid"Işık, Hüseyin"https://zbmath.org/authors/?q=ai:isik.huseyinSummary: The purpose of this paper is to present the notion of MR-Kannan type contractions using the generalized averaged operator. Several examples are provided to illustrate the concept presented herein. We provide a characterisation of normed spaces using MR-Kannan type contractions with a fixed point. We investigate the Ulam-Hyers stability and well-posedness result for the mappings presented here.Some properties and applications of Menger probabilistic inner product spaceshttps://zbmath.org/1522.540432023-12-07T16:00:11.105023Z"Xiao, Jian-Zhong"https://zbmath.org/authors/?q=ai:xiao.jianzhong"Zhu, Xing-Hua"https://zbmath.org/authors/?q=ai:zhu.xinghuaSummary: In this paper a new definition for Menger probabilistic inner product spaces is presented. In the new setting a Menger probabilistic inner product space can naturally become a Menger probabilistic normed space, and a classical inner product space can be considered as a special case of Menger probabilistic inner product spaces. An example is given to illustrate that, the new definition is a nontrivial generalization for classical inner product spaces, and so it has rich contents in probability. By virtue of this definition, the topological structure is discussed and some elementary properties are described in terms of the families of semi-inner products. Also, some convergence theorems and probabilistic Pythagorean theorem are given. As applications, a new fixed point theorem for nonlinear contractions in Menger probabilistic inner product spaces is established.The Mardešić conjecture for countably compact spaceshttps://zbmath.org/1522.540442023-12-07T16:00:11.105023Z"Ishiu, Tetsuya"https://zbmath.org/authors/?q=ai:ishiu.tetsuyaThe Mardešić conjecture states that if \(d\) and \(s\) are positive integers and if \(K_i\) is a compact LOTS for each \(i < d\), \(Z_j\) is an infinite Hausdorff space for each \(j < d + s\), and there exists a continuous surjection from \(\prod_{i<d} K_i \) onto \(\prod_{j<d+s} Z_j\), then there exist at least \( (s + 1)\)-many indices \(j < d + s\) such that \(Z_j\) is metrizable. This conjecture which was proved by \textit{G. Martínez-Cervantes} and \textit{G. Plebanek} [Proc. Am. Math. Soc. 147, No. 4, 1763--1772 (2019; Zbl 1412.54038)] is a generalization of a result of \textit{L. B. Treybig} [ibid. 15, 866--871 (1964; Zbl 0124.15702)] which states that if \(X\) and \(Y\) are infinite Hausdorff spaces and \(X\times Y\) is the continuous image of a compact LOTS, then both \(X\) and \(Y\) are metrizable. The paper under review generalizes the above mentioned result of Martínez-Cervantes and Plebanek, by showing the following: If \(d\) and \(s\) are positive integers, \(K_i\) is a compact LOTS for each \(i < d\), \(X\) is a countably compact subspace of \(\prod_{i<d} K_1\), \(Z_j\) is an infinite Hausdorff space for each \(j < d + s\), and there exists a continuous surjection from \(X \) onto \(\prod_{j<d+s} Z_j\), then there exist at least \( (s + 1)\)-many indices \(j < d + s\) such that \(Z_j\) is compact and metrizable.
Reviewer: Richard G. Wilson (Ciudad de México)\(R^4\)-continua and pseudo-contractibilityhttps://zbmath.org/1522.540452023-12-07T16:00:11.105023Z"Capulín, Félix"https://zbmath.org/authors/?q=ai:capulin.felix"Flores-González, Mario"https://zbmath.org/authors/?q=ai:flores-gonzalez.mario"Maya, David"https://zbmath.org/authors/?q=ai:maya-escudero.david"Orozco-Zitli, Fernando"https://zbmath.org/authors/?q=ai:orozco-zitli.fernandoAuthors' summary: ``A recent result in continuum theory states that each continuum containing an \(R^i\)-set with \(i\in\{1,2,3\}\) is non-pseudo-contractible. In this paper, we introduce the concept of \(R^4\)-continuum and mainly show a new class of non-pseudo-contractible continua, we give the relationship between \(R^4\)-continua and \(R^i\)-sets with \(i\in\{1,2,3\}\), as well as the relationship between \(R^4\)-continua and \(s\)-points, we prove that the standard hyperspaces of a continuum containing an \(R^4\)-continuum also contain an \(R^4\)-continuum, finally, and we prove that if \(f:X\to Y\) is an open map between continua and \(Y\) contains an \(R^4\)-continuum, then \(X\) contains an \(R^4\)-continuum.
Reviewer: Javier Sánchez Martínez (Tuxtla Gutiérez)Countable dense sets of dendriteshttps://zbmath.org/1522.540462023-12-07T16:00:11.105023Z"García-Becerra, Rafael E."https://zbmath.org/authors/?q=ai:garcia-becerra.rafael-e"de J. López, María"https://zbmath.org/authors/?q=ai:de-j-lopez.maria"Pellicer-Covarrubias, Patricia"https://zbmath.org/authors/?q=ai:pellicer-covarrubias.patriciaIn this paper the authors study the countable dense homogeneity degree which is analogous to the homogeneity degree of a topological space. They present a full characterization of the countable homogeneity degree for the class of dendrites. They show for example that for a dendrite, the degree is finite iff it is a tree. And they characterize for this class of spaces when it is countably infinite. In all other cases it is continuum. This interesting paper ends with similar results on dendroids and also states an interesting open problem.
Reviewer: Jan van Mill (Amsterdam)Inverse limits with Smith functionshttps://zbmath.org/1522.540472023-12-07T16:00:11.105023Z"Varagona, Scott"https://zbmath.org/authors/?q=ai:varagona.scottInverse limits on the interval \([0,1]\) with upper semi-continuous (u.s.c.) functions can be described in the following way. Take a sequence \(\{M_{n}\}_{n=1}^{\infty}\) of closed subsets of the square \([0,1]^{2}\) and define the inverse limit \(X\) of the sequence \(\{[0,1],M_{n}\}_{n=1}^{\infty}\) by \(X=\{(x_{1},x_{2},\ldots)\in [0,1]^{\infty}:\) for each \(n\geq 1\), \((x_{n+1},x_{n})\in M_{n}\}\). Since they were introduced by \textit{W. Mahavier} [Topology Appl. 141, No. 1--3, 225--231 (2004; Zbl 1078.54021)], many aspects of these inverse limits have been studied by a number of authors. The freedom on the choice of the sets \(M_{n}\) offers many possibilities, but also restricts the possibility of obtaining general results. For this reason, researchers have restricted their attention to certain specific families of sets such as: continuum-valued (u.s.c.), closed sets which are the union of graphs of mappings, and closed sets which are arcs.
In the paper under review the author studies sets (called by the author \textit{Smith functions}) which are the union of a finite number of convex segments that are either horizontal or vertical. He presents numerous examples, open problems and results concerning connectedness and indecomposability.
Reviewer: Alejandro Illanes (Ciudad de México)Vitali selectors in topological groups and related semigroups of setshttps://zbmath.org/1522.540482023-12-07T16:00:11.105023Z"Chatyrko, Vitalij"https://zbmath.org/authors/?q=ai:chatyrko.vitalij-a"Nyagahakwa, Venuste"https://zbmath.org/authors/?q=ai:nyagahakwa.venuste(no abstract)Countably compact group topologies on arbitrarily large free abelian groupshttps://zbmath.org/1522.540492023-12-07T16:00:11.105023Z"Bellini, Matheus K."https://zbmath.org/authors/?q=ai:bellini.matheus-koveroff"Hart, Klaas Pieter"https://zbmath.org/authors/?q=ai:hart.klaas-pieter"Rodrigues, Vinicius O."https://zbmath.org/authors/?q=ai:rodrigues.vinicius-o"Tomita, Artur H."https://zbmath.org/authors/?q=ai:tomita.artur-hideyukiThe authors address a question first posed by D. Dikranjan and D. Shakhmatov in 1992, concerning the nature of (Hausdorff) group topologies on free abelian groups. They prove that given certain conditions on selective ultrafilters (the existence of continuum many pairwise incompatible selective ultrafilters), a group topology can be established on any free abelian group of cardinality \(\kappa = \kappa^\omega\) such that every finite power of it is countably compact and without non-trivial convergent sequences. This means that countably compact groups may exist for unbounded cardinalities.
Historically, Tomita, Tkachenko, and others made strides in this direction, with Tomita showing that free abelian groups do not admit compact Hausdorff group topologies or even countably compact group topologies for their countable power. Yet, when endowed with certain properties under the Continuum Hypothesis, free abelian groups can have countably compact Hausdorff group topologies. Most known examples have their cardinalities bounded by cardinality of the power-set of the continuum. Building on previous works, the authors show that under the Generalised Continuum Hypothesis, a countably compact group topology (without non-trivial convergent sequences) may be defined on a free abelian group of cardinality \(\kappa\) if and only if \(\kappa = \kappa^\omega\). Moreover, the authors show the consistency of a proper class of cardinals of countable cofinality that can be weights of a countably compact free abelian groups.
Reviewer: Tomasz Kania (Praha)Walrasian auctioneer: an application of Brouwer's fixed point theoremhttps://zbmath.org/1522.540502023-12-07T16:00:11.105023Z"Almuhur, Eman"https://zbmath.org/authors/?q=ai:almuhur.eman"Albalawi, Areej Ali"https://zbmath.org/authors/?q=ai:albalawi.areej-aliSummary: One of this paper's main goals is to demonstrate Brouwer's fixed point theorem which mainly depends on the closed unit ball fixed point condition in \(\mathbb{R}^n\). After that, we demonstrate the Walrasian auctioneer which is an economical application of Brouwer's fixed point theorem where it is a hypothetical technique of price adjustment that mimics how markets reach equilibrium as it increases the price of a good if demand exceeds supply. After price has increased, the Walrasian normalizes the increase. When this new pricing is revealed, the process is then repeated.On new coincidence points results for a mapping and a relation approach to the integral integral equationshttps://zbmath.org/1522.540512023-12-07T16:00:11.105023Z"Ameer, Eskandar"https://zbmath.org/authors/?q=ai:ameer.eskandar"Arshad, Muhammad"https://zbmath.org/authors/?q=ai:arshad.muhammad"Sintunavarat, Wutiphol"https://zbmath.org/authors/?q=ai:sintunavarat.wutipholSummary: In this article, we introduce the new type of contractions for a mapping and a relation and prove certain coincidence point theorems which generalize some known results in this area. Moreover, an application to the existence of a unique solution for the integral equation is also provided.Orbitally \(p\)-implicit \(\mathcal{R}\)-contractive mapping and applications to nonlinear matrix equations and integral equationshttps://zbmath.org/1522.540522023-12-07T16:00:11.105023Z"Bag, Usha"https://zbmath.org/authors/?q=ai:bag.usha"Jain, Reena"https://zbmath.org/authors/?q=ai:jain.reenaSummary: We introduce a \(p\)-implicit \(\mathcal{R}\)-contractive mappings in the context of relational metric spaces with \(w\)-distance and derive fixed points findings from them, followed by two appropriate illustrations. This work is applied to obtain the solution of a nonlinear matrix equation (NME) of the form \(\mathcal{K}=\mathcal{Q}+\sum_{\jmath=1}^k\mathcal{G}_\jmath^\ast\hbar(\mathcal{K})\mathcal{G}_\jmath\), where \(\mathcal{Q}\) is an \(n\times n\) Hermitian positive definite matrix, \(\mathcal{G}_\jmath\) (\(\jmath\in\{1,2,\ldots,m\}\)) are \(n\times n\) matrices, and \(\hbar\) is a non-linear self-mapping of the set of all Hermitian matrices which are continuous in the trace norm. We discuss adequate prerequisites establishing the existence of a unique positive definite solution of the provided NME. An example with a randomly generated matrix, convergence and error analysis, and surface plot for solution visualisation are described. Second, we use our result to provide adequate criteria for the existence of solutions for integral equation generated from fractional differential equations.Existence of best proximity points satisfying two constraint inequalitieshttps://zbmath.org/1522.540532023-12-07T16:00:11.105023Z"Balraj, Duraisamy"https://zbmath.org/authors/?q=ai:balraj.duraisamy"Marudai, Muthaiah"https://zbmath.org/authors/?q=ai:marudai.muthaiah"Mitrovic, Zoran D."https://zbmath.org/authors/?q=ai:mitrovic.zoran-d"Ege, Ozgur"https://zbmath.org/authors/?q=ai:ege.ozgur"Piramanantham, Veeraraghavan"https://zbmath.org/authors/?q=ai:piramanantham.veeraraghavanSummary: In this paper, we prove the existence of best proximity point and coupled best proximity point on metric spaces with partial order for weak proximal contraction mappings such that these critical points satisfy some constraint inequalities.Generic properties of nonexpansive mappings on unbounded domainshttps://zbmath.org/1522.540542023-12-07T16:00:11.105023Z"Bargetz, Christian"https://zbmath.org/authors/?q=ai:bargetz.christian"Reich, Simeon"https://zbmath.org/authors/?q=ai:reich.simeon"Thimm, Daylen"https://zbmath.org/authors/?q=ai:thimm.daylenSummary: We investigate typical properties of nonexpansive mappings on unbounded complete hyperbolic metric spaces. For two families of metrics of uniform convergence on bounded sets, we show that the typical nonexpansive mapping is a Rakotch contraction on every bounded subset and that there is a bounded set which is mapped into itself by this mapping. In particular, we obtain that the typical nonexpansive mapping in this setting has a unique fixed point which can be reached by iterating the mapping. Nevertheless, it turns out that the typical mapping is not a Rakotch contraction on the whole space and that it has the maximal possible Lipschitz constant of one on a residual subset of its domain. By typical we mean that the complement of the set of mappings with this property is \(\sigma\)-\(\phi\)-porous, that is, small in a metric sense. For a metric of pointwise convergence, we show that the set of Rakotch contractions is meagre.Connected \(\epsilon\)-chainable sets and existence resultshttps://zbmath.org/1522.540552023-12-07T16:00:11.105023Z"Bhandari, Samir Kumar"https://zbmath.org/authors/?q=ai:bhandari.samir-kumar"Chandok, Sumit"https://zbmath.org/authors/?q=ai:chandok.sumit"Jana, Bishnupada"https://zbmath.org/authors/?q=ai:jana.bishnupada"Das, Radha Binod"https://zbmath.org/authors/?q=ai:das.radha-binodIn the setting of \(\epsilon\)-chainable metric spaces, the authors introduce \((\epsilon - \rho - \sigma)\) uniformly local weak contractions and obtain some results on the existence of fixed points. Some examples show that the results are satisfied with \(\epsilon\)-chainable metric spaces, but they do not validate in the setting of metric spaces and generalized metric spaces.
Reviewer: Zoran D. Mitrović (Banja Luka)Solution of nonlinear Fredholm integral equation via generalized complex metric spacehttps://zbmath.org/1522.540562023-12-07T16:00:11.105023Z"Gnanaprakasam, Arul Joseph"https://zbmath.org/authors/?q=ai:gnanaprakasam.arul-joseph"Mani, Gunaseelan"https://zbmath.org/authors/?q=ai:mani.gunaseelan"Boulaaras, Salah"https://zbmath.org/authors/?q=ai:boulaaras.salah-mahmoudSummary: In this paper, we introduce a new concept of generalized complex metric spaces and prove fixed point theorems. The presented results generalize and expand some of the literature's well-known results. We also explore some of the applications of our key results.Fixed-point theorems involving Lipschitz in the smallhttps://zbmath.org/1522.540572023-12-07T16:00:11.105023Z"Indrati, Christiana Rini"https://zbmath.org/authors/?q=ai:indrati.christiana-riniSummary: Lipschitz in the small is a generalization of the Lipschitz condition. The Lipschitz condition guarantees the uniqueness of the solution of initial value problems. A~special Lipschitz condition in the small is a contraction in the small. Based on the Lipschitz in the small in this paper, fixed-point theorems involving contraction in the small will be presented. The results will be applied to develop Picard's theorem.Common stationary point of multivalued asymptotically regular mappingshttps://zbmath.org/1522.540582023-12-07T16:00:11.105023Z"Khan, Abdul Rahim"https://zbmath.org/authors/?q=ai:khan.abdul-rahim"Oyetunbi, Dolapo Muhammed"https://zbmath.org/authors/?q=ai:oyetunbi.dolapo-muhammed"Izuchukwu, Chinedu"https://zbmath.org/authors/?q=ai:izuchukwu.chineduSummary: We establish a relationship between asymptotic regularity and common stationary points of multivalued mappings on a metric space. As a consequence of our results, we obtain a new common fixed point result for two asymptotically regular single-valued mappings. Our work significantly improves and complements comparable results in the literature.Fuzzy Kakutani-Fan-Glicksberg fixed point theorem and existence of Nash equilibria for fuzzy gameshttps://zbmath.org/1522.540592023-12-07T16:00:11.105023Z"Liu, Jiuqiang"https://zbmath.org/authors/?q=ai:liu.jiuqiang"Yu, Guihai"https://zbmath.org/authors/?q=ai:yu.guihaiSummary: In this paper, we give fuzzy generalizations to the well-known Brouwer fixed point theorem and Kakutani-Fan-Glicksberg fixed point theorem. As applications, we apply the fuzzy Kakutani-Fan-Glicksberg fixed point theorem to derive an existence theorem for Nash equilibria in generalized fuzzy games with locally convex Hausdorff topological vector spaces for strategy spaces and/or discontinuous payoff functions which generalizes existence theorems for Nash equilibria in generalized games.Fixed point results via real-valued function satisfying integral type rational contractionhttps://zbmath.org/1522.540602023-12-07T16:00:11.105023Z"Mani, Naveen"https://zbmath.org/authors/?q=ai:mani.naveen"Sharma, Amit"https://zbmath.org/authors/?q=ai:sharma.amit-raj|sharma.amit-kumar"Shukla, Rahul"https://zbmath.org/authors/?q=ai:shukla.rahul(no abstract)Generalized Hutchinson operator in \(G\)-metric spaces via generalized iterated function systemhttps://zbmath.org/1522.540612023-12-07T16:00:11.105023Z"Nazir, Talat"https://zbmath.org/authors/?q=ai:nazir.talatSummary: Existence of common attractors of \(G\)-iterated function system are established in \(G\)-metric spaces. Consequently, we acquire different results for \(G\)-iterated function systems based of finite family of general contractive conditions. Some examples are reinforce the results proved herein. The well-posedness of attractor-based problems of generalized Hutchinson contractive operators in the framework of Hausdorff \(G\)-metric spaces is also established.A new fixed point theorem in \(G_b\)-metric space and its application to solve a class of nonlinear matrix equationshttps://zbmath.org/1522.540622023-12-07T16:00:11.105023Z"Pakhira, Samik"https://zbmath.org/authors/?q=ai:pakhira.samik"Hossein, Sk Monowar"https://zbmath.org/authors/?q=ai:hossein.sk-monowarSummary: In this article we extend the notions of \(G\)-metric and \(b\)-metric and define a new metric called \(G_b\)-metric with coefficient \(b\ge 1\). A fixed point theorem is proved in this metric space. We obtain parallel results of several existing fixed point theorems such as that of Banach, Geraghty and Boyd-Wong in \(G_b\)-metric space using our theorem. As an application of our fixed point theorem we provide a fixed point iteration to solve a class of nonlinear matrix equations of the form \(X^s+A^*G(X)A=Q\), where \(s\ge 1\), \(A\) is an \(n\times n\) matrix, \(G\) is a continuous function from the set of all Hermitian positive definite matrices to the set of all Hermitian positive semi-definite matrices and \(Q\) is an \(n\times n\) Hermitian positive definite matrix. It is noted that the error in estimated solution we get by following our method is lesser than the error we get with Ćirić's fixed point iteration.Graphical contractions and common fixed points in \(b\)-metric spaceshttps://zbmath.org/1522.540632023-12-07T16:00:11.105023Z"Petruşel, Adrian"https://zbmath.org/authors/?q=ai:petrusel.adrian"Petruşel, Gabriela"https://zbmath.org/authors/?q=ai:petrusel.gabrielaSummary: In this paper, we will prove some fixed point theorems for graphical contractions in complete \(b\)-metric spaces. Then, some common fixed point results for a pair of mappings in complete \(b\)-metric spaces will be deduced. Our results extend some recent theorems proved in classical metric spaces.On a pair of fuzzy dominated mappings on closed ball in the multiplicative metric space with applicationshttps://zbmath.org/1522.540642023-12-07T16:00:11.105023Z"Rasham, Tahair"https://zbmath.org/authors/?q=ai:rasham.tahair"Shabbir, Muhammad Sajjad"https://zbmath.org/authors/?q=ai:shabbir.muhammad-sajjad"Agarwal, Praveen"https://zbmath.org/authors/?q=ai:agarwal.praveen"Momani, Shaher"https://zbmath.org/authors/?q=ai:momani.shaher-mSummary: The purpose of this paper is to establish some fixed point results for a pair of fuzzy dominated mappings satisfying contractive conditions on closed ball in multiplicative metric space. Some new fixed point results with graphic contraction on closed ball for a pair of fuzzy graph dominated mappings on multiplicative metric space have been established. Furthermore, we find a unique common solution for a system of non linear Voltera type integral equations and lastly we give an application to ensure the existence of common bounded solution of a functional equation in dynamic programming.\(\delta\)-convex structure on rectangular metric spaces concerning Kannan-type contraction and Reich-type contractionhttps://zbmath.org/1522.540652023-12-07T16:00:11.105023Z"Sharma, Dileep Kumar"https://zbmath.org/authors/?q=ai:sharma.dileep-kumar"Tiwari, Jayesh"https://zbmath.org/authors/?q=ai:tiwari.jayesh-kSummary: In the present paper, we introduce the notation of \(\delta\)-convex rectangular metric spaces with the help of convex structure. We investigate fixed point results concerning Kannan-type contraction and Reich-type contraction in such spaces. We also propound an ingenious example in reference of given new notion.Coincidence fixed-point theorems for \(p\)-hybrid contraction mappings in \(G_b\)-metric space with applicationhttps://zbmath.org/1522.540662023-12-07T16:00:11.105023Z"Wangwe, Lucas"https://zbmath.org/authors/?q=ai:wangwe.lucas(no abstract)A homological version of a result of digital topologyhttps://zbmath.org/1522.540672023-12-07T16:00:11.105023Z"Kong, T. Yung"https://zbmath.org/authors/?q=ai:kong.tat-yungLet \(\mathcal{F}\) be a collection that satisfies conditions (1) and (2) (see Page 353) of Section 1 of the present paper and \(\mathcal{D} \subset \mathcal{F}\). Then the paper gives a local characterization of the members of \(\mathcal{D}\) that are \(P\)-homology-simple for \(\mathcal{D}\) in \(\mathcal{F}\). Besides, it also generalizes this result. Furthermore, the result of the current paper can be compared with Bertrand and Couprie's results in [\textit{G. Bertrand} and \textit{M. Couprie}, J. Math. Imaging Vis. 35, No. 1, 23--35 (2009; Zbl 1490.68260); ibid. 31, 35--56 (2008; \url{doi:10.1007/s10851-007-0063-0}); ibid. 48, No. 1, 134--148 (2014; Zbl 1291.68397)].
Reviewer: Sang-Eon Han (Jeonju)On a factorized \(L\)-fuzzy automaton and its \(L\)-fuzzy topological characterizationhttps://zbmath.org/1522.682852023-12-07T16:00:11.105023Z"Singh, Shailendra"https://zbmath.org/authors/?q=ai:singh.shailendra|singh.shailendra-narayan|singh.shailendra-kumar.1|singh.shailendra-kumar"Tiwari, S. P."https://zbmath.org/authors/?q=ai:tiwari.s-p"Pal, Priyanka"https://zbmath.org/authors/?q=ai:pal.priyankaSummary: This paper is towards the study of \(L\)-fuzzy automata via theory of factorizations, where \(L\) is a complete residuated lattice. We study the concept of factorization of \(L\)-fuzzy sets and factorization of collection of \(L\)-fuzzy sets. Such factorizations are used (i) to study the concept of minimal \(L\)-fuzzy automaton, and (ii) to introduce \(L\)-fuzzy topologies for the study of concepts associated with an \(L\)-fuzzy automaton. Specifically, we introduce the notion of factorized \(L\)-fuzzy automaton corresponding to a given \(L\)-fuzzy automaton and study the minimality of such automaton. Further, we associate \(L\)-fuzzy topologies for factorized \(L\)-fuzzy automaton and use these to characterize some algebraic concepts.Introduction to nonextensive statistical mechanics. Approaching a complex worldhttps://zbmath.org/1522.820022023-12-07T16:00:11.105023Z"Tsallis, Constantino"https://zbmath.org/authors/?q=ai:tsallis.constantinoNonextensive statistical mechanics was introduced by the author in [\textit{C. Tsallis}, J. Stat. Phys. 52, No. 1--2, 479--487 (1988; Zbl 1082.82501)] as a possible generalization of Boltzmann-Gibbs statistical mechanics with the goal to enlarge the class of systems allowing for a statistical mechanical description. Roughly speaking, the basic mathematical idea is to replace throughout the exponential function \(e^x\) (which is a solution to \(\frac{dy}{dx} = y\)) by the \(q\)-exponential function:
\[
e_q^x := \big(1 + (1-q)x\big)_+^{1/(1-q)} \quad (e_1^x := e^x)
\]
(which solves \(\frac{dy}{dx}=y^q\)) and the natural logarithm \(\ln x\) by the \(q\)-logarithmic function:
\[
\ln_q x := \frac{1-x^{1-q}}{q-1} \quad (\ln_1 x := \ln x)
\]
(which is the inverse of the \(q\)-exponential function). In particular, the nonextensive statistical mechanics is built on the \(q\)-generalization of the Boltzmann-Gibbs entropy, where the natural logarithm is replaced by the \(q\)-logarithm (\(q\neq 1\)):
\[
S_q := k \int dx\, p(x)\, \ln_q \big(1/p(x)\big) = k \frac{1 - \int dx\, [p(x)]^q}{q-1}.
\]
(The BG-entropy is recovered in the limit \(q\to 1\).)
This book provides an introduction into the foundations of nonextensive statistical mechanics and an overview of its numerous applications. In the first part of the book (Chapters 1--3), the entropy \(S_q\) is introduced and its properties are proven. In particular, it is shown that \(S_q\) is nonadditive and extensive. In the second part (Chapters 4--6), the aspects of nonextensive statistical mechanics are developed in the context of nonlinear dynamical systems, at mesoscopic level (e.g., the Langevin and the Fokker-Planck equations) as well as at microscopic level (e.g., dissipative maps, conservative maps, many-body long-range-interacting Hamiltonian systems). Some \(q\)-generalizations of the central limit theorem and the large deviations theory are also discussed. The third part (Chapter 7) contains an overview of concrete and typical applications of the nonextensive framework in physics, chemistry, economics, computer science, bioscience, cellular automata, self-organized criticality, scale-free networks, linguistics. In the fourth part (Chapter 8), conjectures and open problems of the theory are discussed as well as recent progress towards their verification. Finally, the author answers to some common critiques of the theory in the form of question-answer, which also serves a good summary of the key ideas developed throughout the book.
Reviewer: Artem Sapozhnikov (Leipzig)Random lift of set valued maps and applications to multiagent dynamicshttps://zbmath.org/1522.910402023-12-07T16:00:11.105023Z"Capuani, Rossana"https://zbmath.org/authors/?q=ai:capuani.rossana"Marigonda, Antonio"https://zbmath.org/authors/?q=ai:marigonda.antonio"Ricciardi, Michele"https://zbmath.org/authors/?q=ai:ricciardi.micheleSummary: We introduce an abstract framework for the study of general mean field games and mean field control problems. Given a multiagent system, its macroscopic description is provided by a time-depending probability measure, where at every instant of time the measure of a set represents the fraction of (microscopic) agents contained in it. The trajectories available to each of the microscopic agents are affected also by the overall state of the system. By using a suitable concept of random lift of set valued maps, together with fixed point arguments, we are able to derive properties of the macroscopic description of the system from properties of the set valued map expressing the admissible trajectories for the microscopical agents. The techniques used can be applied to consider a broad class of dependence between the trajectories of the single agent and the state of the system. We apply the results in the case in which the admissible trajectories of the agents are the minimizers of a suitable integral functional depending also from the macroscopic evolution of the system.Topological obstructions to stability and stabilization. History, recent advances and open problemshttps://zbmath.org/1522.930052023-12-07T16:00:11.105023Z"Jongeneel, Wouter"https://zbmath.org/authors/?q=ai:jongeneel.wouter"Moulay, Emmanuel"https://zbmath.org/authors/?q=ai:moulay.emmanuelPublisher's description: This open access book provides a unified overview of topological obstructions to the stability and stabilization of dynamical systems defined on manifolds and an overview that is self-contained and accessible to the control-oriented graduate student. The authors review the interplay between the topology of an attractor, its domain of attraction, and the underlying manifold that is supposed to contain these sets. They present some proofs of known results in order to highlight assumptions and to develop extensions, and they provide new results showcasing the most effective methods to cope with these obstructions to stability and stabilization. Moreover, the book shows how Borsuk's retraction theory and the index-theoretic methodology of Krasnosel'skii and Zabreiko underlie a large fraction of currently known results. This point of view reveals important open problems, and for that reason, this book is of interest to any researcher in control, dynamical systems, topology, or related fields.