Recent zbMATH articles in MSC 54https://zbmath.org/atom/cc/542023-01-20T17:58:23.823708ZWerkzeugTree-like constructions in topology and modal logichttps://zbmath.org/1500.030052023-01-20T17:58:23.823708Z"Bezhanishvili, G."https://zbmath.org/authors/?q=ai:bezhanishvili.guram"Bezhanishvili, N."https://zbmath.org/authors/?q=ai:bezhanishvili.nick"Lucero-Bryan, J."https://zbmath.org/authors/?q=ai:lucero-bryan.joel-gregory"van Mill, J."https://zbmath.org/authors/?q=ai:van-mill.janIn topological semantics, the modal language is interpreted in a topological space \(X\) by evaluating propositional variables as subsets of \(X\), classical connectives as Boolean operations on the powerset of \(X\), \(\square\) as the interior operator and \(\lozenge\) as the closure operator. A formula \(\varphi\) is valid in \(X\), denoted \(X \models \varphi\), provided \(\varphi\) evaluates to \(X\) under any evaluation of the propositional variables. The logic of \(X\) is \(\mathsf{Log}(X) := \{\varphi \mid X \models \varphi\}\), and one has \(\mathsf{S4} \subseteq \mathsf{Log}(X)\).
Many topological completeness results have been obtained since the creation of topological semantics. The main purpose of this paper is to provide a uniform approach to topological completeness results in modal logic for zero-dimensional Hausdorff spaces. The authors get it by developing a general technique for topologizing trees with limits, within ZFC. The technique also allows the authors to obtain new topological completeness results for non-metrizable spaces. By embedding these spaces into well-known extremally disconnected spaces then the authors also get new completeness results for logics extending \(\mathsf{S4.2}\).
Specifically, the unified way of obtaining a zero-dimensional topology on an infinite tree \(T\) with limits is by designating a particular Boolean algebra of subsets of \(T\) as a basis. If \(T\) has countable branching, then the topology ends up being metrizable. If the branching is 1, then the obtained space is homeomorphic to the ordinal space \(\omega + 1\); if the branching is \(\ge 2\) but finite, then it is homeomorphic to the Pełczyński compactification of \(\omega\); and if the branching is countably infinite, then there are subspaces homeomorphic to the space of rational numbers, the Baire space, as well as to the ordinal spaces \(\omega^n +1\). Mapping theorems for these countable branching trees with limits lead to alternate proofs of some well-known topological completeness results for \(\mathsf{S4}\), \(\mathsf{S4.1}\), \(\mathsf{Grz}\), and \(\mathsf{Grz}_n\) with respect to zero-dimensional metrizable spaces.
For uncountable branching, it is required to designate a Boolean \(\sigma\)-algebra as a basis for the topology. This leads to topological completeness results for \(\mathsf{S4}\), \(\mathsf{S4.1}\), \(\mathsf{Grz}\), and \(\mathsf{Grz}_n\) with respect to non-metrizable zero-dimensional Hausdorff spaces.
To obtain the topological completeness results for logics extending \(\mathsf{S4.2}\), the authors select a dense subspace of either the Čech-Stone compactification \(\beta D\) of a discrete space \(D\) with large cardinality or the Gleason cover \(E\) of a large enough power of \([0, 1]\). This selection is realized by embedding a subspace of an uncountable branching tree with limits into either \(\beta D\) or \(E\). The latter gives rise to \(\mathsf{S4.2}\), while the former yields the other logics of interest extending \(\mathsf{S4.2}\).
Reviewer: Jorge Picado (Coimbra)On equicontinuous factors of flows on locally path-connected compact spaceshttps://zbmath.org/1500.370122023-01-20T17:58:23.823708Z"Edeko, Nikolai"https://zbmath.org/authors/?q=ai:edeko.nikolaiSuppose that \((K,T)\) and \((L,T)\) are flows in the topological dynamics settings. Then \((L,T)\) is called a factor of \((K,T)\) if there exists a continuous surjective map \(p:K\to L\) such that \(p(tx)=tp(x)\) for all \(t\in T\). A flow \((L,T)\) is called a maximal equicontinuous factor of \((K,T)\) if it is an equicontinuous factor of \((K,T)\) with the additional property that every other equicontinuous factor of \((K,T)\) is also a factor of \((L,T)\).
The main theorem in the paper is a characterization of an equicontinuous factor.
Theorem. Let \((K,G)\) be a flow such that \(K\) is locally path-connected with finite first Betti number \(b_1(K)\), \(G\) is abelian, and \(K\) is metrizable or \(G\) is separable. If all \(G\)-invariant functions \(f\in C(K)\) are constant, then every equicontinuous factor of \((K,G)\) is isomorphic to a minimal flow on some compact abelian Lie group of dimension less than \(b_1(K)/b_0(K)\).
An important ingredient in the proof is the following result by \textit{T. Hauser} and \textit{T. Jäger} [Proc. Am. Math. Soc. 147, No. 10, 4539--4554 (2019; Zbl 1431.37040); Theorem 2.12]:
Theorem. For a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone.
Now, let \(X\) and \(Y\) be topological spaces and \(p:X\to Y\) be a map. Then \(p\) is called monotone if for each \(y\in Y\) the preimage \(p^{-1}(y)\) is a connected subset of \(X\). The author provides a new proof of the above theorem. The proof uses a new characterization of the monotonicity of a quotient map \(p:K\to L\) thanks to some properties of the Banach lattice of complex continuous functions \(C(K)\) in the case of locally connected compact spaces \(K\) and \(L\).
Another result used in the proof of the main theorem is taken from [\textit{J. S. Calcut} et al., Topology Appl. 159, No. 1, 322--330 (2012; Zbl 1233.54003); Theorem 1.1]. This claim the following.
Theorem. Let \(f:(X,x_0)\to (Y,y_0)\) be a quotient map of pointed topological spaces, where \(X\) is locally path-connected and \(Y\) is semilocally simply-connected. If each fiber \(f^{-1}(y)\) is connected, then the induced homomorphism \(f_*:\pi_1(X,x_0)\to \pi_1(Y,y_0)\) of the fundamental groups is surjective.
Reviewer: Nikita Shekutkovski (Skopje)Equicontinuity and regionally proximal relation via Furstenberg familieshttps://zbmath.org/1500.370142023-01-20T17:58:23.823708Z"Ri, SongHun"https://zbmath.org/authors/?q=ai:ri.songhun"Ju, HyonHui"https://zbmath.org/authors/?q=ai:ju.hyonhui"Kim, CholSan"https://zbmath.org/authors/?q=ai:kim.cholsanIn this paper, a maximal \(\mathcal{F}\)-equicontinuous factor is given for some topological dynamical systems. Then, an equivalence result is established between \(\mathcal{F}\)-equicontinuity and \(\mathcal{F}\)-distality of induced systems on hyperspaces.
The following are the main results.
Theorem. Let \((X,T)\) be a topological dynamical system
and \(\mathcal{F}\) be a filter. Then, the following claims hold true:
(1) If \((X,T)\) is \(\mathcal{F}\)-equicontinuous then \((X,T)\) is \(\mathcal{F}\)-distal.
(2) \((X,T)\) is \(\mathcal{F}\)-equicontinuous if and only if \(Q_{k,\mathcal{F}}(X,T)=\Delta_X\) (here \(Q_{k,\mathcal{F}}(X,T)\) is the set of all
\(\mathcal{F}\)-regionally proximal pairs of \((X,T)\) and \(\Delta_X = \{(x, x) \in X \times X : x \in X\}\)).
Theorem. Let \((X,T)\) and \((Y,S)\) be a pair of topological dynamical systems and \(\mathcal{F}\) be a filter. Further, let \(\pi:X\to Y\) be an open map. Then \((Y,S)\) is \(\mathcal{F}\)-equicontinuous if and only if
\(Q_{k,\mathcal{F}}(X,T)\subset R_\pi\) (here \(R_\pi = \{(x,y\} \in X \times X : T(x)= T(y)\)).
Further aspects involving the \(\mathcal{F}\)-equicontinuous characterization of topological dynamical systems with a thick filter are also discussed.
Reviewer: Mihai Turinici (Iaşi)Double Wijsman asymptotic \(\mathcal{I}_2\)-invariant equivalencehttps://zbmath.org/1500.400032023-01-20T17:58:23.823708Z"Dündar, Erdinç"https://zbmath.org/authors/?q=ai:dundar.erdinc"Akın, Nimet Pancaroğlu"https://zbmath.org/authors/?q=ai:akin.nimet-pancaroglu"Ulusu, Uğur"https://zbmath.org/authors/?q=ai:ulusu.ugurSummary: In this study, for double set sequences, we present the notions of Wijsman asymptotic invariant equivalence, Wijsman asymptotic \(\mathcal{I}_2\)-invariant equivalence, and Wijsman asymptotic \(\mathcal{I}^*_2\)-invariant equivalence. Also, we examine the relations between these notions and Wijsman asymptotic invariant statistical equivalence studied in this field before.Continuous linear images of spaces \(C_p(X)\) with the weak topologyhttps://zbmath.org/1500.460032023-01-20T17:58:23.823708Z"Kąkol, Jerzy"https://zbmath.org/authors/?q=ai:kakol.jerzy"Leiderman, Arkady"https://zbmath.org/authors/?q=ai:leiderman.arkady-gSummary: \(C_p(X)\) denotes the space of continuous real-valued functions on a Tychonoff space \(X\) with the topology of pointwise convergence. A locally convex space (lcs) \(E\) with the weak topology is denoted by \(E_w\). First, we show that there is no a sequentially continuous linear surjection \(T: C_p(X)\rightarrow E_w\), if \(E\) is a lcs with a fundamental sequence of bounded sets. Second, we prove that if there exists a sequentially continuous linear map from \(C_p(X)\) onto \(E_w\) for some infinite-dimensional metrizable lcs \(E\), then the completion of \(E\) is isomorphic to the countable power of the real line \(\mathbb{R}^\omega\). Illustrating examples are provided.Continuity of extensions of Lipschitz mapshttps://zbmath.org/1500.460592023-01-20T17:58:23.823708Z"Ciosmak, Krzysztof J."https://zbmath.org/authors/?q=ai:ciosmak.krzysztof-jSummary: We establish the sharp rate of continuity of extensions of \(\mathbb{R}^m\)-valued 1-Lipschitz maps from a subset \(A\) of \(\mathbb{R}^n\) to a 1-Lipschitz maps on \(\mathbb{R}^n\). We consider several cases when there exists a 1-Lipschitz extension with preserved uniform distance to a given 1-Lipschitz map. We prove that, if \(m > 1\), then a given map is 1-Lipschitz and affine if and only if such a distance preserving extension exists for any 1-Lipschitz map defined on any subset of \(\mathbb{R}^n\). This shows a striking difference from the case \(m = 1\), where any 1-Lipschitz function has such a property. Another example where we prove it is possible to find an extension with the same Lipschitz constant and the same uniform distance to another Lipschitz map \(v\) is when the difference between the two maps takes values in a fixed one-dimensional subspace of \(\mathbb{R}^m\) and the set \(A\) is geodesically convex with respect to a Riemannian pseudo-metric associated with \(v\).A note on the topological transversality theorem for maps with lower semicontinuous type selectionshttps://zbmath.org/1500.470722023-01-20T17:58:23.823708Z"O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donalSummary: This paper presents a Leray-Schauder alternative and a topological transversality theorem for compact maps which have lower semicontinuous type selections. Basically if we have two compact maps \(F\) and \(G\) which have lower semicontinuous type selections and \(F\cong G\) (defined in an approriate way) then one map being essential guarantees the essentiality of the other map.A fixed point theorem and an application for the Cauchy problem in the scale of Banach spaceshttps://zbmath.org/1500.470832023-01-20T17:58:23.823708Z"Vo Viet Tri"https://zbmath.org/authors/?q=ai:vo-viet-tri."Karapinar, Erdal"https://zbmath.org/authors/?q=ai:karapinar.erdalSummary: The main aim of this paper is to prove the existence of the fixed point of the sum of two operators in setting of the cone-normed spaces with the values of cone-norm belonging to an ordered locally convex space. We apply this result to prove the existence of global solution of the Cauchy problem with perturbation of the form
\[
\begin{cases}
x^\prime(t) = f [t, x (t)] + g[t, x (t)], \; t\in [0,\infty), \\
x(0) = x_0 \in F_1,
\end{cases}
\]
in a scale of Banach spaces \( \{(F_s, \|.\|_s) : s \in (0,1] \} \).Some convergence theorems for monotone nonexpansive mappings in hyperbolic metric spaceshttps://zbmath.org/1500.471152023-01-20T17:58:23.823708Z"Uddin, Izhar"https://zbmath.org/authors/?q=ai:uddin.izhar"Aggarwal, Sajan"https://zbmath.org/authors/?q=ai:aggarwal.sajan"Muangchoo-In, Khanitin"https://zbmath.org/authors/?q=ai:muangchoo-in.khanitin"Kumam, Poom"https://zbmath.org/authors/?q=ai:kumam.poomSummary: In this paper, we prove strong and \(\Delta\)-convergence theorem for monotone nonexpansive mapping in a partially ordered hyperbolic metric space using Mann iteration scheme. Moreover, we give an numerical example to illustrate the main result in this paper.General metric spaces warped along submetryhttps://zbmath.org/1500.530592023-01-20T17:58:23.823708Z"Walczak, Szymon"https://zbmath.org/authors/?q=ai:walczak.szymon-mIn this paper, the author generalizes the concept of warped product to warped metric along submetry between two compact geodesic general metric spaces. The study provides the necessary and sufficient conditions for the convergence of the sequences of warped general metric spaces in the generalized Gromov-Hausdorff topology.
Reviewer: Surabhi Tiwari (Allahabad)\(T_4\), Urysohn's lemma, and Tietze extension theorem for constant filter convergence spaceshttps://zbmath.org/1500.540012023-01-20T17:58:23.823708Z"Baran, Tesnim Meryem"https://zbmath.org/authors/?q=ai:baran.tesnim-meryem"Erciyes, Ayhan"https://zbmath.org/authors/?q=ai:erciyes.ayhanThe two contributors of this work are working in the field of filter convergence spaces, which were studied and examined by various authors in the past. Their main focus lies in the characterization of each of the various forms of local \(T_4\) constant filter convergence spaces and stating the relationships among these various forms. It is shown that the subcategories of local \(T_4\) constant filter convergence spaces are productive and hereditary. And at the end of the paper the authors present a generalization of both the classical Urysohn Lemma and Tietze Extension Theorem but now in the realm of constant filter convergence spaces.
In this context a filter convergence space consists on a pair \((B,K)\), where \(B\) is a non-empty set and \(K\) a map from \(B\) into the power set \(P(F(B))\) of filters on \(B\), satisfying the following conditions: For each \(x\in B\) the point filter \([\{x\}]\) is an element of \(K(x)\), and filters finer than filters in \(K(x)\) also belong to \(K(x)\). Then continuous mappings between filter convergence spaces are defined in an obvious way. The authors denote by \textbf{FCO} the category of filter convergence spaces and continuous maps, and by \textbf{ConFCO} the subcategory of \textbf{FCO} whose objects are pairs \((B,K)\), where \(K\) is a constant function. This concept goes back to \textit{F. Schwarz} [Lect. Notes Math. 719, 345--357 (1979; Zbl 0409.54002)], who proved that \textbf{ConFCO} and the category \textbf{FIL} of filter spaces and continuous maps are isomorphic. Moreover, it is well-known that \textbf{FIL} and \textbf{FMER} the category of filter merotopic spaces and corresponding morphisms in the sense of Kate'tov are isomorphic, too. Since \textbf{FMER} constitutes a quasitopos, \textbf{ConFCO} also possesses this desirable property.
Reviewer: Dieter Leseberg (Berlin)On points of convergence lattices and sobriety for convergence spaceshttps://zbmath.org/1500.540022023-01-20T17:58:23.823708Z"Mynard, Frédéric"https://zbmath.org/authors/?q=ai:mynard.fredericIn this paper, the author studies convergence spaces \((X,\xi)\) such that the space of points of \((\mathbb{P}X,\lim_\xi)\) in the category of convergence lattices is \((X,\xi)\). Especially, the author studies conditions on a convergence space \((X,\xi)\) under which ultrafilters are principal. For that, properties of convergence spaces are studied. The meanings of sober, weakly sober, quasi-order, weakly quasi-order, antisymmetric and almost antisymmetric in the related environment of convergence spaces are investigated. The author makes a special research for the convergence space \(\mathrm{pt}L\), where \((L,\lim_L)\) is a convergence lattice, and the space \(\mathrm{pt}(\mathbb{P}X,\lim_\xi)\). Also, the \(\mathcal{Z}\)-regularity, sobrification and \(T_D\) axiom in convergence spaces have an important role in the paper. Many examples complete this study.
Reviewer: Dimitrios Georgiou (Patras)On the functor of probability measures and quantization dimensionshttps://zbmath.org/1500.540032023-01-20T17:58:23.823708Z"Ivanov, A. V."https://zbmath.org/authors/?q=ai:ivanov.aleksandr-vladimirovich.1|ivanov.aleksandr-vladimirovich|ivanov.aleksandr-valentinovichSummary: The quantization dimensions of the probability measure given on the metric compact coincide with the dimensions of the finite approximation for the probability measure functor. Some functorial properties of quantization dimensions are established. It is shown that for any \(b>0\) there exists a metric compact \(X_b\) of capacitive dimension \(\text{dim}_{\text{B}}X_b = b\) on which there are probability measures with support equal to \(X\) whose quantization dimension takes all possible values from the interval \([0, b]\).Topologies generated by symmetric porosity on normed spaceshttps://zbmath.org/1500.540042023-01-20T17:58:23.823708Z"Kowalczyk, Stanisław"https://zbmath.org/authors/?q=ai:kowalczyk.stanislaw"Turowska, Małgorzata"https://zbmath.org/authors/?q=ai:turowska.malgorzataIn this manuscript the authors consider families of symmetrically porouscontinuous functions. In the first part of the paper maximal additive families for symmetrically porouscontinuous functions are investigated. The main part of this section is a technical Theorem 2.6. It consists of a number of equivalent conditions concerning sums of \(P_r^s\)-continuous, \(S_r^s\)-continuous and \(M_r^s\)-continuous functions. The conclusion from this theorem is that the maximal additive class for the families of functions that are \(P_r^s\)-continuous or \(S_r^s\)-continuous or \(M_r^s\)-continuous, is the family of continuous mappings. In the second part of the paper maximal multiplicative families for symmetrically porouscontinuous functions are investigated. The main part of this section are Theorems 3.7 and 3.8. They provide conditions equivalent for the function \(f\) to belong to a maximal multiplicative family for \(M_r^s\)-continuous and \(S_r^s\)-continuous functions, respectively. In the last part of the paper some properties of topologies generated by porosity are given.
Reviewer: Waldemar Sieg (Bydgoszcz)Zeros of functionals and a parametric version of Michael selection theoremhttps://zbmath.org/1500.540052023-01-20T17:58:23.823708Z"Fomenko, T. N."https://zbmath.org/authors/?q=ai:fomenko.tatyana-nikolaevnaThe author suggests a modification of her method of search for zeros of functionals (see, e.g., [\textit{T. N. Fomenko}, Math. Notes 93, No. 1, 172--186 (2013; Zbl 1273.54053); translation from Mat. Zametki 93, No. 1, 127--143 (2013)]) and presents on that base a result on the existence, for all values of the parameter \(t\in [0;1]\), of single-valued continuous selections for a given parametric family \(F_t\) of multimaps with closed values acting from a topological space \(X\) to a complete metric space \((Y,d)\) provided such selection exists for \(F_0.\) It should be noticed that the mentioned selections are located in a given open subset of the metric space \((C(X,Y),\mu)\) of single-valued continuous maps from \(X\) to \(Y\), with the metric \(\mu(f,g):=\sup\limits_{x\in X}\,d(f(x),g(x))\).
As a consequence of this assertion, the author obtains the statement, representing a parametric version of the Michael continuous selection theorem for a family of lower semicontinuous multimaps with closed convex values.
Reviewer: Valerii V. Obukhovskij (Voronezh)The generalization of ``Aleksandrov's dublicate'' of Sorgenfrey line and set rational pointshttps://zbmath.org/1500.540062023-01-20T17:58:23.823708Z"Khmylëva, T. E."https://zbmath.org/authors/?q=ai:khmyleva.tatyana-evgenievna|khmyleva.tatiana-evgenievna|khmyleva.tatana-eSummary: In this paper we consider the generalization of wellknown ``Aleksandrov's dublicate''. It is proved, that \(X\times n\) and \(X\times m\) are not homeomorphic for Sorgenfrey line \(X\) and for different integers \(n, m\). If \(X\) is the rationals, then \(X\otimes n\) and \(X\otimes m\) are homeomorphic for all integers \(n\) and \(m\).Semi-Kelley compactifications of \((0,1]\)https://zbmath.org/1500.540072023-01-20T17:58:23.823708Z"Chacón-Tirado, Mauricio"https://zbmath.org/authors/?q=ai:chacon-tirado.mauricio-esteban"Embarcadero-Ruiz, Daniel"https://zbmath.org/authors/?q=ai:embarcadero-ruiz.daniel"Naranjo-Murillo, Jimmy A."https://zbmath.org/authors/?q=ai:naranjo-murillo.jimmy-a"Vidal-Escobar, Ivon"https://zbmath.org/authors/?q=ai:vidal-escobar.ivonA continuum \(X\) is said to be semi-Kelley provided that for each subcontinuum \(K\) and for any two maximal limit continua \(M\) and \(L\) in \(K\) either \(M\subset L\) or \(L\subset M\). A Kelley continuum \(X\) is a Kelley compactification if it is a compactification of \((0, 1]\). A semi-Kelley continuum \(X\) is a semi-Kelley compactification if it is a compactification of \((0, 1]\). A continuum \(X\) is a Kelley remainder, respectively semi-Kelley remainder, if it is the remainder of a Kelley compactification, respectively semi-Kelley compactification.
The purpose of this paper under review is to characterize the semi-Kelley compactifications of \((0, 1]\) with remainder being an arc or a simple closed curve. At the end of this article the authors also pose the following question: Let \(X\) be an arc-like or circle-like semi-Kelley continuum. Is \(X\) a semi-Kelley remainder?
Reviewer: Alicia Santiago Santos (Oaxaca)On the uniqueness of the \(n\)-fold pseudo-hyperspace suspension for locally connected continuahttps://zbmath.org/1500.540082023-01-20T17:58:23.823708Z"Libreros-López, Antonio"https://zbmath.org/authors/?q=ai:libreros-lopez.antonio"Macías-Romero, Fernando"https://zbmath.org/authors/?q=ai:macias-romero.fernando"Herrera-Carrasco, David"https://zbmath.org/authors/?q=ai:herrera-carrasco.davidA continuum is a compact connected nondegenerate metric space. By a hyperspace of a continuum \(X\) we mean a specified collection of subsets of \(X\) endowed with the Hausdorff metric. Given a continuum and \(n\in \mathbb N\), we consider its hyperspaces:
\[
\begin{aligned}
2^{X} &= \{A\subseteq X \mid A \text{ is closed and nonempty}\},\\
\mathcal{C}_{n}(X) &= \{A\in 2^{X}\mid A \text{ has at most \(n\) components}\}, \text{ and}\\
\mathcal{F}_{n}(X) &= \{A\in 2^{X}\mid A \text{ has at most \(n\) points}\}.
\end{aligned}
\]
We topologize these sets with the Hausdorff metric. \textit{J. C. Macías} [Glas. Mat., III. Ser. 43, No. 2, 439--449 (2008; Zbl 1160.54005)] introduced the notion of the \textit{\(n\)-fold pseudo-hyperspace suspension} of a continuum \(X\) denoted by \(\mathcal{PHS}_n(X)\), and defined it as the quotient space:
\[
\mathcal{PHS}_n(X)=\mathcal{C}_n(X)/ \mathcal{F}_1(X),
\]
topologized with the quotient topology.
On the other hand, given a continuum \(X\), let \(\mathcal{H}(X)\) be any of the hyperspaces \(2^X, \mathcal{C}_{n}(X), \mathcal{F}_{n}(X)\). The continuum \(X\) is said to have unique hyperspace \(\mathcal{H}(X)\) provided that the following implication holds: if \(Y\) is a continuum and \(\mathcal{H}(X)\) is homeomorphic to \(\mathcal{H}(Y)\), then \(X\) is homeomorphic to \(Y\).
The purpose of the paper under review is to prove that meshed continua have unique \(n\)-fold pseudo-hyperspace suspension, for \(n > 1\).
The authors also pose the following open question: Let \(X\) be a locally connected continuum such that \(X\) is not almost meshed and let \(n \in \mathbb{N}\). Does \(X\) have unique hyperspace \(\mathcal{PHS}_n(X)\)?
Reviewer: Alicia Santiago Santos (Oaxaca)Cardinal invariants of coset spaceshttps://zbmath.org/1500.540092023-01-20T17:58:23.823708Z"Ling, Xuewei"https://zbmath.org/authors/?q=ai:ling.xuewei"He, Wei"https://zbmath.org/authors/?q=ai:he.wei.3|he.wei.2|he.wei|he.wei.1"Lin, Shou"https://zbmath.org/authors/?q=ai:lin.shouAuthors' abstract: A topological space \(X\) is called a \textit{coset} space if \(X\) is homeomorphic to a quotient space \(G/H\) of left cosets, for some closed subgroup \(H\) of a topological group \(G\). In this paper, we investigate the cardinal invariants of coset spaces. We first show that if \(H\) is a closed neutral subgroup of a topological group \(G\), then \(\Delta(G/H) =\psi(G/H)\), \(w(G/H) =d(G/H)\cdot \chi(G/H)\) and \(w(G/H) =l(G/H) \cdot \chi(G/H)\).
We also prove that if \(H\) is a closed subgroup of a feathered topological group \(G\), then (1) \(w(G/H) =d(G/H) \cdot\chi(G/H)\) and \(w(G/H) =l(G/H)\cdot\chi(G/H)\); (2) the quotient space \(G/H\) is metrizable if and only if \(G/H\) is first-countable.
At the end, we consider some applications of \(sp\)-networks in coset spaces. In particular, we show that if \(H\) is a closed neutral subgroup of a topological group \(G\), then (1) \(spnw(G/H) =d(G/H) \cdot sp_\chi(G/H)\); (2) the quotient space \(G/H\) is metrizable if and only if \(G/H\) has countable \(sp\)-character.
Reviewer: Ivan S. Gotchev (New Britain)Fixed point theorem for multivalued non-self mappings satisfying JS-contraction with an applicationhttps://zbmath.org/1500.540102023-01-20T17:58:23.823708Z"Aron, David"https://zbmath.org/authors/?q=ai:aron.david"Kumar, Santosh"https://zbmath.org/authors/?q=ai:kumar.santosh|kumar.santosh.2|kumar.santosh.1|kumar.santosh.4|kumar.santosh.3Summary: In this paper, we present some fixed point results for multivalued non-self mappings. We generalize the fixed point theorem due to \textit{I. Altun} and \textit{G. Minak} [Carpathian J. Math. 32, No. 2, 147--155 (2016; Zbl 1399.54079)] by using \textit{M. Jleli} and \textit{B. Samet}'s [J. Inequal. Appl. 2014, Paper No. 38, 8 p. (2014; Zbl 1322.47052)] \( \vartheta \)-contraction. To validate the results proved here, we provide an appropriate application of our main result.Revisiting Meir-Keeler type fixed operators on Branciari distance spacehttps://zbmath.org/1500.540112023-01-20T17:58:23.823708Z"Fulga, Andreea"https://zbmath.org/authors/?q=ai:fulga.andreea"Karapinar, Erdal"https://zbmath.org/authors/?q=ai:karapinar.erdalIn this paper, the authors prove some Meir-Keeler type fixed point theorems in the setting of Branciari distance space. An example is also given in support of their proved result. The results of the paper improve and extend a host of previously known results and are indeed useful to the researchers working in nonlinear analysis, particularly, in the area of metric fixed point theory.
Reviewer: Ravindra Kishor Bisht (Pune)Common fixed point theorems for expansive type mappings in cone heptagonal metric spaceshttps://zbmath.org/1500.540122023-01-20T17:58:23.823708Z"Haile, Gizachew"https://zbmath.org/authors/?q=ai:haile.gizachew"Koyas, Kidane"https://zbmath.org/authors/?q=ai:koyas.kidane"Girma, Aynalem"https://zbmath.org/authors/?q=ai:girma.aynalemSummary: In this paper, we establish common fixed point theorems for a pair of self-mappings involving expansive type conditions in cone heptagonal metric spaces. The proved results generalize, improve and extend some known results in the literature. We also provide examples in support of our main results.Characterizations of manifolds modeled on absorbing sets in non-separable Hilbert spaces and the discrete cells propertyhttps://zbmath.org/1500.570212023-01-20T17:58:23.823708Z"Koshino, Katsuhisa"https://zbmath.org/authors/?q=ai:koshino.katsuhisaThe author states in the abstract: ``We characterize infinite-dimensional manifolds modeled on absorbing sets in non-separable Hilbert spaces by using the discrete cells property, which is a general position property. Moreover, we study the discrete (locally finite) approximation property, which is an extension of the discrete cells property.'' Let us make a few remarks and also state one of the results.
A reader of this paper or even this review should have available \textit{K. Sakai}'s volume [Topology of infinite-dimensional manifolds. Cham: Springer (2020; Zbl 1481.57002)] in order to comprehend the terms used herein as we shall not define them. Let us mention however, in passing, that we do not understand the definition of \(\mathfrak{C}_\sigma\) after the first bullet in the Introduction. What does it mean for a space \(A_n\in\mathfrak{C}\) to be closed? The sets defined apparently are just unions of topological spaces, indexed by \(n\in\omega\), so they would not necessarily be topological spaces.
The notions of \(Z\)-set, strong \(Z\)-set, \(Z\)-embedding, and \(\mathfrak{C}\)-absorbing set are reviewed in the Introduction. Let us state one of the results.
\textbf{Theorem 1.1.} Let \(\mathfrak{C}\) be a topological and closed hereditary class, and \(\Omega\) be a \(\mathfrak{C}\)-absorbing set in \(\ell_2(\kappa)\). For a connected space \(X\in\mathfrak{C}_\sigma\) of density \(\leq\kappa\), the following are equivalent:
\begin{itemize}
\item \(X\) is an \(\Omega\)-manifold.
\item \(X\) can be embedded into some \(\ell_2(\kappa)\)-manifold as a \(\mathfrak{C}\)-absorbing set.
\item \(X\) satisfies the following conditions:
\begin{itemize}
\item[(a)] \(X\) is an \(\mathrm{ANR};\)
\item[(b)] \(X\) is strongly \(\mathfrak{C}\)-universal;
\item[(c)] \(X\) has the discrete \(n\)-cells property for every \(n\in\omega\);
\item[(d)] \(X\) is a strong \(Z_\sigma\)-set in itself.
\end{itemize}
\end{itemize}
Theorems 1.2 and 1.3 are stated on page 129. The first has to do with recognizing when a certain space is an absorbing set, whereas, the second is a characterization of a connected space \(X\) in a certain class being an \((\ell_2^f\times\Omega)\)-manifold.
Reviewer: Leonard R. Rubin (Norman)