Recent zbMATH articles in MSC 54A25https://zbmath.org/atom/cc/54A252021-06-15T18:09:00+00:00WerkzeugOn certain versions of straightness.https://zbmath.org/1460.540182021-06-15T18:09:00+00:00"Das, Pratulananda"https://zbmath.org/authors/?q=ai:das.pratulananda"Pal, Sudip Kumar"https://zbmath.org/authors/?q=ai:pal.sudip-kumar"Adhikary, Nayan"https://zbmath.org/authors/?q=ai:adhikary.nayanAll spaces in the sequel are metric and \(C(X)\) denotes the set of all continuous real-valued functions on \(X\). \textit{A. Berarducci} et al. [Topology Appl. 146--147, 339--352 (2005; Zbl 1078.54014)] introduced and studied the notion of straight spaces: A space \(X\) is \textit{straight} if whenever \(X\) is the union of two closed sets, then \(f\in C(X)\) is uniformly continuous iff its restriction to each of the closed sets is uniformly continuous.
In this paper, the authors consider notions of continuity which are strictly weaker than the notion of uniform continuity, and introduce three versions of straightness, namely pre-straightness, pre\((\ast)\)-straightness and \(W\)-straightness: A space \(X\) is \textit{pre-straight} if whenever \(X\) is the union of closed sets, then \(f\in C(X)\) is Cauchy regular iff its restriction to each of the closed sets is Cauchy regular, where a mapping \(f:X\to Y\) is Cauchy regular if for any Cauchy sequence \(\{x_n\}\) in \(X\), \(\{f(x_n)\}\) is Cauchy in \(Y\). A space \(X\) is \textit{pre\((\ast)\)-straight} if whenever \(X\) is the union of closed sets, then a real-valued Cauchy regular function \(f\) is uniformly continuous iff its restriction to each of the closed sets is uniformly continuous. Recall that a sequence \(\{x_n\}\) in a metric space \((X, d)\) is \textit{quasi-Cauchy} if given \(\varepsilon>0\), there is a natural number \(n_0\) such that \(d(x_{n+1},x_n)<\varepsilon\) for all \(n\geq n_0\). A space \(X\) is \textit{\(W\)-straight} if whenever \(X\) is the union of closed sets, then \(f\in C(X)\) is ward continuous iff its restriction to each closed sets is ward continuous, where a mapping \(f: X\to Y\) is \textit{ward continuous} if it preserves quasi-Cauchy sequences.
The authors investigate some properties of these spaces and the relationship between these notions. For instance, they obtain the following: A space \(X\) is straight if and only if \(X\) is pre-straight and pre\((\ast)\)-straight. A space \(X\) is pre\((\ast)\)-straight if and only if its completion \(\widehat{X}\) is straight.
Reviewer: Kohzo Yamada (Shizuoka)Star versions of the Rothberger property on hyperspaces.https://zbmath.org/1460.540072021-06-15T18:09:00+00:00"Casas-de la Rosa, Javier"https://zbmath.org/authors/?q=ai:casas-de-la-rosa.javier"Martínez-Ruiz, Iván"https://zbmath.org/authors/?q=ai:martinez-ruiz.ivan"Ramírez-Páramo, Alejandro"https://zbmath.org/authors/?q=ai:ramirez-paramo.alejandroA space \(X\) is said to be star-Rothberger (resp., strongly star-Rothberger) if for each sequence \((\mathcal U_n)\) of open covers of \(X\) there is a sequence \((U_n)\) (resp., \((x_n)\)) such that for each \(n\), \(U_n \in \mathcal U_n\) (resp., \(x_n \in X\)) and \(\{St(U_n,\mathcal U_n): n \in \mathbb N\}\) (resp., \(\{St(x_n,\mathcal U_n): n \in \mathbb N\}\)) is an open cover of \(X\). The authors characterize the star-Rothberger and strongly star-Rothberger properties of the hyperspace over a space \(X\) equipped with the Fell topology in terms of selective properties of \(X\) of \(\pi\)-network-type. They also characterize the strongly star-Rothberger property on hyperspaces endowed with the lower Vietoris topology. Selective versions of metacompactness and and mesocompactness are also considered.
Reviewer: Ljubiša D. Kočinac (Niš)Rothberger and Rothberger-type star selection principles on hyperspaces.https://zbmath.org/1460.540082021-06-15T18:09:00+00:00"Díaz-Reyes, Jesús"https://zbmath.org/authors/?q=ai:reyes.jesus-diaz"Ramírez-Páramo, Alejandro"https://zbmath.org/authors/?q=ai:ramirez-paramo.alejandro"Tenorio, Jesús F."https://zbmath.org/authors/?q=ai:tenorio.jesus-fFor a topological space \(X\), let \(CL (X)\) denote the hyperspace consisting of all nonempty closed subsets of \(X\) endowed with the Vietoris topology, and let \(\mathbb{K}(X)\) (resp., \(\mathbb{F}(X)\), \(\mathbb{CS}(X)\)) be the subspace consisting of all nonempty compact subsets (resp., nonempty finite subsets, convergent sequences) of \(X\). Motivated by the work of \textit{Z. Li} [Topology Appl. 212, 90--104 (2016; Zbl 1355.54014)], the authors characterize the Rothberger property and two selection principles called star-Rothberger and strongly star-Rothberger in the spaces \(CL (X)\), \(\mathbb{K}(X)\), \(\mathbb{F}(X)\) and \(\mathbb{CS}(X)\).
Let \(X\) be a topological space and let \(\Lambda\) be one of the hyperspaces \(CL (X)\), \(\mathbb{K}(X)\), \(\mathbb{F}(X)\) and \(\mathbb{CS}(X)\). Let \(\mathbf{S}_1(\mathcal{A},\mathcal{B})\) be the selection principle defined by \textit{M. Scheepers} [Topology Appl. 69, No. 1, 31--62 (1996; Zbl 0848.54018)]. The authors introduce the notion of \(\pi_V(\Lambda)\)-network modifying the notion of \(\pi_V\)-network defined by Li [loc. cit.], and two selection principles \(\mathbf{S}_{\Pi_V} (\Pi_V(\Lambda),\Pi_V(\Lambda))\) and \(\mathbf{S}^*_{\Pi_V} (\Pi_V(\Lambda),\Pi_V(\Lambda))\), where \(\Pi_V (\Lambda)\) is the collection of \(\pi_V(\Lambda)\)-networks of \(X\). The following theorems are proved: \(\Lambda\) has the Rothberger property if and only if \(X\) satisfies \(\mathbf{S}_{1} (\Pi_V(\Lambda),\Pi_V(\Lambda))\); \(\Lambda\) is strongly star-Rothberger if and only if \(X\) satisfies \(\mathbf{S}_{\Pi_V} (\Pi_V(\Lambda),\Pi_V(\Lambda))\); and \(\Lambda\) is star-Rothberger if and only if \(X\) satisfies \(\mathbf{S}^*_{\Pi_V} (\Pi_V(\Lambda),\Pi_V(\Lambda))\). Let \(\mathcal{D}\) denote the family of dense subsets of a given space. The authors also give a characterization of the selection principle \(\mathbf{S}_1 (\mathcal{D},\mathcal{D})\) for \(\Lambda\) following ideas by Li [loc. cit.].
Reviewer: Takamitsu Yamauchi (Matsuyama)