Recent zbMATH articles in MSC 54Bhttps://zbmath.org/atom/cc/54B2023-11-13T18:48:18.785376ZWerkzeugGeneralized spaces for constructive algebrahttps://zbmath.org/1521.032452023-11-13T18:48:18.785376Z"Blechschmidt, Ingo"https://zbmath.org/authors/?q=ai:blechschmidt.ingoThe text refers to sheaves (locales), geometric theories, sheaf semantics and constructive commutative algebra within the constructive impredicative metatheory which implies that in the presented material, the law of excluded middle is not used, any version of the axiom of choice is not used, and, also, any non-classical logical principles are not added. It consists of three sections. In the second section, the author explores the basics of the theory of locales (sheaves), with a focus on examples and occurrences related to constructive mathematics. Sheaves interact particularly well with geometric theories, which are reviewed in subsection 3.1. Further on, sheaves enable the generalization of the author's notion of a model of geometric theory. Secondly, geometric theories make it possible to efficiently construct the spaces (locales) on which sheaves are to be considered. This is because a fundamental feature of geometric theories (in contrast to arbitrary first-order theories, for example) is that their models naturally organize (Subsection 3.2). Sheaves are the subject of Subsections 3.3 and 3.4. Each of the mentioned sections ends with a large number of very interesting examples.
For the entire collection see [Zbl 1486.03009].
Reviewer: Daniel A. Romano (Banja Luka)Isometric embeddings of bounded metric spaces in the Gromov-Hausdorff classhttps://zbmath.org/1521.510052023-11-13T18:48:18.785376Z"Ivanov, Alexandr O."https://zbmath.org/authors/?q=ai:ivanov.aleksandr-olegovich"Tuzhilin, Alexey A."https://zbmath.org/authors/?q=ai:tuzhilin.alexey-aThe authors show that any bounded metric space can be embedded isometrically in the Gromov-Hausdorff metric class \(\mathcal{GH}\) (the class of isometry classes of all metric spaces, endowed with the Gromov-Hausdorff distance). This follows from the description of the local geometry of \(\mathcal{GH}\) in a sufficiently small neighbourhood of a generic metric space.
Reviewer: Victor V. Pambuccian (Glendale)Some topological and cardinal properties of the \(\lambda_\tau^\varphi \)-nucleus of a topological space \(X \)https://zbmath.org/1521.540022023-11-13T18:48:18.785376Z"Yuldashev, T. K."https://zbmath.org/authors/?q=ai:yuldashev.tursun-kamaldinovich"Mukhamadiev, F. G."https://zbmath.org/authors/?q=ai:mukhamadiev.farkhod-gSummary: In this paper, we study the behavior of some topological and cardinal properties of topological spaces under the influence of the \(\lambda_{\tau}^{\varphi}\)-kernel of a topological space \(X\). It has been proved that the \(\lambda_{\tau}^{\varphi}\)-kernel of a topological space \(X\) preserves the density and the network \(\pi \)-weight of normal spaces.The Vietoris functor and modal operators on rings of continuous functionshttps://zbmath.org/1521.540042023-11-13T18:48:18.785376Z"Bezhanishvili, G."https://zbmath.org/authors/?q=ai:bezhanishvili.guram"Carai, L."https://zbmath.org/authors/?q=ai:carai.luca"Morandi, P. J."https://zbmath.org/authors/?q=ai:morandi.patrick-jIn this paper, the authors introduce an endofunctor \(\mathcal{H}\) on the category \(\textbf{bal}\) of bounded archimedean \(\textit{l}\)-algebras and show that there is a dual adjunction between the category \(\textbf{Alg}(\mathcal{H})\) of algebras for \(\mathcal{H}\) and the category \(\textbf{Coalg}(\mathcal{V})\) of coalgebras for the Vietoris endofunctor \(\mathcal{V}\) on the category of compact Hausdorff spaces. They also introduce an endofunctor \(\mathcal{H}^u\) on the reflective subcategory of \(\textbf{bal}\) consisting of uniformly complete objects of \(\textbf{bal}\) and show that Gelfand duality lifts to a dual equivalence between \(\textbf{Alg}(\mathcal{H}^u)\) and \(\textbf{Coalg}(\mathcal{V})\). Finally, some known results are generalized.
Reviewer: Sami Lazaar (Sidi Daoued)The Xi-Zhao model of \(T_1\)-spaceshttps://zbmath.org/1521.540052023-11-13T18:48:18.785376Z"Chen, Siheng"https://zbmath.org/authors/?q=ai:chen.siheng"Li, Qingguo"https://zbmath.org/authors/?q=ai:li.qingguoIn this paper, for a \(T_1\) space \(X\), the authors study properties of the dcpo model \(D(X)\) constructed by \textit{D. Zhao} and \textit{X. Xi} [Math. Proc. Camb. Philos. Soc. 164, No. 1, 125--134 (2018; Zbl 1469.06012)]. They prove the following results:
\begin{itemize}
\item[(1)] \(X\) is weak well-filtered if and only if \(D(X)\) is weak well-filtered;
\item[(2)] \(D(X)\) is open well-filtered;
\item[(3)] \(D(X)\) is core-compact if and only if it is locally compact if and only if it is quasicontinuous.
\item[(4)] The special Xi-Zhao dcpo model \(\widehat{Zh(X)}\) of a \(T_1\) space \(X\) is coherent if and only if every subset of \(X\) is compact.
\end{itemize}
Throughout the paper there are examples and questions that enrich the study.
Reviewer: Dimitrios Georgiou (Pátra)Inverse limit slender groupshttps://zbmath.org/1521.540062023-11-13T18:48:18.785376Z"Conner, Gregory R."https://zbmath.org/authors/?q=ai:conner.gregory-r"Herfort, Wolfgang"https://zbmath.org/authors/?q=ai:herfort.wolfgang-n"Kent, Curtis"https://zbmath.org/authors/?q=ai:kent.curtis-a"Pavešić, Petar"https://zbmath.org/authors/?q=ai:pavesic.petarAuthors' summary: Classically, an abelian group \(G\) is said to be slender if every homomorphism from the countable product \(\mathbb{Z}^{\mathbb{N}}\) to \(G\) factors through the projection to some finite product \(\mathbb{Z}^n\). Various authors (e.g. [\textit{G. R. Conner} and \textit{S. M. Corson}, Proc. Am. Math. Soc. 147, No. 3, 1255--1268 (2019; Zbl 1477.20049); \textit{K. Eda}, J. Algebra 148, No. 1, 243--263 (1992; Zbl 0779.20012)]) have proposed generalizations to non-commutative groups; this has resulted in a plethora of similar but not completely equivalent concepts. In the first part of this work we present a unified treatment of these concepts and examine how
they are related. In the second part we study slender groups in the context of \textit{co-small} objects in certain categories, and give several new applications, including the proof that certain homology groups of Barratt-Milnor spaces are cotorsion groups, and a universal coefficient theorem for Čech cohomology with coefficients in a slender group.
Reviewer: Tayyebe Nasri (Bojnord)On localization of the Menger propertyhttps://zbmath.org/1521.540102023-11-13T18:48:18.785376Z"Alam, Nur"https://zbmath.org/authors/?q=ai:alam.nur"Chandra, Debraj"https://zbmath.org/authors/?q=ai:chandra.debrajSummary: In this paper we introduce and study the local version of the Menger property, namely locally Menger property (or, locally Menger space). We explore some preservation like properties in this space. We also discuss certain situations where this local property behaves somewhat differently from the classical Menger property. Some observations about the character of a point, network weight and weight in this space are also investigated carefully. We also introduce the notion of Menger generated space (in short, MG-space) and make certain investigations in these spaces. Several topological observations on the decomposition and the remainder of locally Menger spaces are also discussed.Products of sequentially compact spaces with no separability assumptionhttps://zbmath.org/1521.540122023-11-13T18:48:18.785376Z"Lipparini, Paolo"https://zbmath.org/authors/?q=ai:lipparini.paoloRecall that \(\mathfrak s\) denotes the splitting number, and \(\mathfrak h\) denotes the distributivity number. As noted by the author, using Booth's Theorem 2 in [\textit{D. Booth}, Fundam. Math. 85, 99--102 (1974; Zbl 0297.54020)], or van Douwen's Theorem 6.1 in [\textit{K. Kunen} (ed.) and \textit{J. E. Vaughan} (ed.), Handbook of set-theoretic topology. Amsterdam-New York-Oxford: North-Holland. 111--167 (1984; Zbl 0546.00022)], one may define \(\mathfrak s\) to be the smallest cardinal such that \(2^{\mathfrak s}\) is not sequentially compact, and using results of \textit{P. Simon} [Rend. Ist. Mat. Univ. Trieste 25, No. 1--2, 447--450 (1993; Zbl 0839.54002)], define \(\mathfrak h\) to be the smallest cardinal such that there are \(\mathfrak h\) sequentially compact spaces whose product is not sequentially compact. In some of the above references all spaces were assumed to be Hausdorff or stronger.
In the article being reviewed, all spaces are assumed to be nonempty, and as suggested in the title, no separation axioms are assumed unless stated otherwise. A space \(X\) is called \textit{ultraconnected} if no pair of nonempty closed subsets of \(X\) is disjoint. Some of the results obtained are these:
Theorem 2.4. A product of topological spaces is sequentially compact if and only if all of its subproducts by \(\leq{\mathfrak s}\) factors are sequentially compact. The author notes that 2.4 is primarily of interest with no separation axioms assumed, since for \(T_1\)-spaces the result is immediate from the definitions.
Theorem 2.6. If \(\mathfrak h=\mathfrak s\), then the following are equivalent for a product \(X\) of topological spaces: (i) \(X\) is sequentially compact; (ii) all factors of \(X\) are sequentially compact, and the set of factors with a nonconverging sequence has cardinality \(<\mathfrak s\); (iii) all factors of \(X\) are sequentially compact, and all but fewer than \(\mathfrak s\) factors are ultraconnected. A proof is sketched that the following are equivalent: \(\mathfrak h=\mathfrak s\); for every product space \(X\) in 2.6, (i) and (ii) are equivalent; for every product space \(X\) with \(\mathfrak h\) factors in 2.6, (ii) implies (i).
Recalling the recent very nice proof by \textit{A. Dow} and \textit{S. Shelah} [Indag. Math., New Ser. 29, No. 1, 382--395 (2018; Zbl 1436.03260)] of the consistency that \(\mathfrak s\) is singular, and noting that several proofs of Blass have shown \(\mathrm{cf}{\mathfrak s}\geq{\mathfrak h}\) (see [\textit{A. Blass}, Contemp. Math. 302, 33--48 (2002; Zbl 1013.03054)]), the author proceeds to present a simple topological proof of that inequality.
In addition, the author discusses some related topics and questions and an approach to studying them using partial infinitary semigroups.
Reviewer: Robert M. Stephenson Jr. (Columbia)The hyperspace of a semi-Eberlein compact space is semi-Eberleinhttps://zbmath.org/1521.540132023-11-13T18:48:18.785376Z"Rojas-Hernández, R."https://zbmath.org/authors/?q=ai:rojas-hernandez.reynaldo"Tenorio, J. F."https://zbmath.org/authors/?q=ai:tenorio.jesus-f"Yescas-Aparicio, C."https://zbmath.org/authors/?q=ai:yescas-aparicio.cLet $T$ be a set. Denote by $c_0([0,1]^T)$ the subspace of $[0,1]^T$ consisting of functions from $T$ to $[0,1]$ taking only a finite subset of $T$ to the half-open interval $(\varepsilon,1]$, for each $\varepsilon>0$. Similarly, denote by $\Sigma[0,1]^T$ the subspace of $[0,1]^T$ (called the \textbf{$\Sigma$-product} of $[0,1]^T$) consisting of functions $f:T\rightarrow [0,1]$ mapping at most a countable number of points of $T$ to the half-open interval $(0,1]$.
A compact space $X$ is \textbf{Eberlein} if, for some set $T$, it has a homeomorphic embedding $f:X\rightarrow c_0([0,1]^T)$, i.e., $X$ is homeomorphic to a subspace of $c_0([0,1]^T)$. A compact space $X$ is \textbf{semi-Eberlein} if, for some set $T$, it has a homeomorphic embedding $f:X\rightarrow [0,1]^T$ such that the preimage $f^{-1}(c_0([0,1]^T))$ is dense in $X$. A compact space $X$ is \textbf{Corson} if, for some set $T$, it has a homeomorphic embedding $f:X\rightarrow\Sigma[0,1]^T$. A compact space $X$ is \textbf{Valdivia} if, for some set $T$, it has a homeomorphic embedding $f:X\rightarrow[0,1]^T$ such that the preimage $f^{-1}(\Sigma[0,1]^T)$ is dense in $X$.
Given a space $X$, let $K(X)$ denote the \textbf{(hyper)space} of compact subsets of $X$ as a metric space with respect to the Hausdorff distance and let $C_p(X)$ denote the space of real-valued continuous functions on $X$ with respect to the topology of poinwise convergence (i.e., the standard product topology on $\mathbb{R}^X$). Let $\omega$ denote the first infinite ordinal and let $\omega_1$ denote the first uncountable ordinal as a space with respect to the standard \textbf{order topology} (the topology generated by intervals of the form $(a,b)=\{x\in\omega_1:a<x<b\}$, for $a,b\in\omega_1$).
This paper (i) establishes that if $X$ is a semi-Eberlein compact space then so is $K(X)$, in particular, answering an open question, Question 1.3, from [\textit{C. Islas} and \textit{D. Jardon}, Open Math. 13, 188--195 (2015; Zbl 1331.54008)]. Other major results of this paper include (ii) characterization of the transfer of certain structures between $X$ and $C_p(X)$ (answering an open question, Question 5.9, from [\textit{F. Casarrubias-Segura} et al., J. Math. Anal. Appl. 451, No. 2, 1154--1164 (2017; Zbl 1397.54035)]) and (iii) characterization of metrizable subspaces of $\omega_1$ (answering an open question, Question 4.7, from [\textit{V. V. Tkachuk}, Topology Appl. 209, 289--300 (2016; Zbl 1346.54004)]).
{\flushleft \textbf{A note from the authors:}} According to the first sentence on page 9 of this paper, it was proved in [Casarrubias-Segura et al., loc. cit.] that ``\textit{$X$ admits a (full) $c$-skeleton if and only if $C_p(X)$ admits a (full) $q$-skeleton}''. Indeed, by [loc. cit., Proposition 5.4], ``\textit{if $X$ admits a (full) $c$-skeleton then $C_p(X)$ admits a (full) $q$-skeleton}''. Conversely, by [loc. cit., Proposition 5.3], ``\textit{if $X$ admits a (full) $q$-skeleton then $C_p(X)$ admits a (full) $c$-skeleton}''. Applying this to $C_p(X)$ we get ``\textit{if $C_p(X)$ admits a (full) $q$-skeleton then $C_p(C_p(X))$ admits a (full) $c$-skeleton}''. By a comment after [loc. cit., Definition 5.2], subspaces inherit $c$-skeletons and so, since $X$ can be embedded in $C_p(C_p(X))$, it follows that ``\textit{if $C_p(X)$ admits a (full) $q$-skeleton then $X$ admits a (full) $c$-skeleton}''.
Reviewer: Earnest Akofor (Bambili)Finite graphs have unique \(n\)-fold symmetric product suspensionhttps://zbmath.org/1521.540172023-11-13T18:48:18.785376Z"Montero-Rodríguez, Germán"https://zbmath.org/authors/?q=ai:montero-rodriguez.german"Herrera-Carrasco, David"https://zbmath.org/authors/?q=ai:herrera-carrasco.david"de J. López, María"https://zbmath.org/authors/?q=ai:de-j-lopez.maria"Macías-Romero, Fernando"https://zbmath.org/authors/?q=ai:macias-romero.fernandoA continuum is a nonempty compact, connected metric space. A finite graph is a continuum that can be written as the union of finitely many arcs, each two of which are either disjoint or intersect only at their end points. For a continuum \(Z\) and \(n\in\mathbb{N}\), let \(F_n(Z)\) be the space of all nonempty closed subsets of \(Z\) which have at most \(n\) points, equipped with the Hausdorff metric. The space \(SF_n(Z)=F_n(Z)/F_1(Z)\) is called the \(n\)-fold symmetric product suspension of \(Z\), and it was defined in [\textit{F. Barragán}, Topology Appl. 157, No. 3, 597--604 (2010; Zbl 1185.54007)]. In this paper the authors prove that if \(X\) is a finite graph, \(n\geq 4\), and \(Y\) is a continuum such that \(SF_n(X)\) and \(SF_n(Y)\) are homeomorphic, then \(Y\) is homeomorphic to \(X\). That is, a finite graph has a unique \(n\)-fold symmetric product suspension for \(n\geq 4\).
Reviewer: Ana Anušić (North Bay)Strongly topological gyrogroups and quotient with respect to \(L\)-subgyrogroupshttps://zbmath.org/1521.540212023-11-13T18:48:18.785376Z"Bao, Meng"https://zbmath.org/authors/?q=ai:bao.meng"Ling, Xuewei"https://zbmath.org/authors/?q=ai:ling.xuewei"Xu, Xiaoquan"https://zbmath.org/authors/?q=ai:xu.xiaoquanThe authors investigate some generalized metric properties in strongly topological gyrogroups. In particular, they prove that when \(G\) is a strongly topological gyrogroup with a symmetric neighborhood base \(U\) at \(0\) and \(H\) is a second-countable admissible subgyrogroup generated from \(U\), if the quotient space \(G/H\) is an \(\aleph_0\)-space (resp., cosmic space), then \(G\) is also an \(\aleph_0\)-space (resp., cosmic space). If the quotient space \(G/H\) has a star-countable \(cs\)-network (resp., \(wcs*\)-network, \(k\)-network), then \(G\) also has a star-countable \(cs\)-network (resp., \(wcs*\)-network, \(k\)-network). Moreover, it is shown that when \(G\) is a strongly topological gyrogroup with a symmetric neighborhood base \(U\) at \(0\) and \(H\) is a locally compact metrizable admissible subgyrogroup generated from \(U\), if the quotient space \(G/H\) is sequential, then \(G\) is also sequential. Furthermore, if the quotient space \(G/H\) is strictly (strongly) Fréchet-Urysohn, then \(G\) is also strictly (strongly) Fréchet-Urysohn. Finally, if the quotient space \(G/H\) is a stratifiable space (semi-stratifiable space, \(\sigma\)-space, \(k\)-semistratifiable space), then \(G\) is a local stratifiable space (semi-stratifiable space, \(\sigma\)-space, \(k\)-semistratifiable space).
Reviewer: Watchareepan Atiponrat (Chiang Mai)\(\mathbb{R}\)-factorizability of topological groups and \(G\)-spaceshttps://zbmath.org/1521.540222023-11-13T18:48:18.785376Z"Martyanov, E. V."https://zbmath.org/authors/?q=ai:martyanov.evgenii-vyacheslavovichThe reviewed paper is (probably) written for researchers with a clear inclination to consider axiomatic theories with roots in general topology, in particular for researchers who want to specialize in some categorical aspects related to topological groups. The considerations concern lots of concepts and facts with which the author only briefly acquaints the readers, referring to the literature not only in the cited monographs. The meaning of some abbreviations must be guessed by the reader, for example an abbreviation ``the category \textbf{G-Tych}'' used more than ten times. For these reasons it is tedious to read, so instead of the author's discussion, I am posting a slightly modified abstract of the paper.
It is investigated the relation between \(\mathbb R\)-factorizability of a \textit{G-}space in the category \textbf{G-Tych} and \(\mathbb R\)-factorizability of its acting group. It is shown that \(\mathbb R\)-factorizability of an acting group with \textit{d-}open action doesn't imply the \(\mathbb R\)-factorizability of a \textit{G-}space. Transitivity of a \textit{d}-open action of an \(\mathbb R\)-factorizable group implies \(\mathbb R\)-factorizability of \textit{G}-space in the category \textbf{G-Tych}. Moreover, if a \textit{d}-openly acting group is openly factorizable, then the \textit{G}-space is \(\mathbb R\)-factorizable in the category \textbf{G-Tych}. Also, a \(\sigma\)-lattice of open homomorphism on \textit{G} induces a \(\sigma\)-lattice of equivariant \textit{d}-open maps on \((G,X,\alpha)\) and \(X\) is an I-favorable space.
Reviewer: Szymon Plewik (Katowice)