Recent zbMATH articles in MSC 54B20https://zbmath.org/atom/cc/54B202021-06-15T18:09:00+00:00WerkzeugA more balanced approach to ideal variation of \(\gamma\)-covers.https://zbmath.org/1460.540132021-06-15T18:09:00+00:00"Das, Pratulananda"https://zbmath.org/authors/?q=ai:das.pratulananda"Samanta, Upasana"https://zbmath.org/authors/?q=ai:samanta.upasana"Chandra, Debraj"https://zbmath.org/authors/?q=ai:chandra.debrajIn this note, the authors introduce the notion of \(G\)-\(\mathcal{I}\)-\(\gamma\) cover as a generalized and more balanced version of \(\mathcal{I}\)-\(\gamma\)-cover and study some of its basic selection properties as also certain Ramsey like properties and splittability properties. Certain characterizations of \(\gamma\)-sets are obtained in terms of \(G\)-\(\mathcal{I}\)-\(\gamma\) covers. The main results are the following:
\textbf{Theorem 1.} If \(X\) satisfies \(S_{fin}(\Omega, \mathcal{O}^{I-gp})\) then \(X\) satisfies \(U_{fin}(\mathcal{O}, G\textmd{-}\mathcal{I}\textmd{-}\Gamma)\) provided \(S_{1}(\mathcal{F}(I), \mathcal{F}(I))\) holds.
\textbf{Theorem 2.} For a space \(X\), \((\begin{array}{l} \ \ \ \Omega\\
G\textmd{-}\mathcal{I}\textmd{-}\Gamma \end{array})=S_{1}(\Omega, G\textmd{-}\mathcal{I}\textmd{-}\Gamma)\).
\textbf{Theorem 3.} For a space \(X\), \(S_{1}(\Omega, \Gamma)=S_{1}(G\textmd{-}\mathcal{I}\textmd{-}\Gamma, \Gamma)\).
Reviewer: Zuquan Li (Hangzhou)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)Borel complexity up to the equivalence.https://zbmath.org/1460.540242021-06-15T18:09:00+00:00"Bartoš, Adam"https://zbmath.org/authors/?q=ai:bartos.adamThe families \(\mathcal F\) of subsets of the hyperspace \(\mathcal K([0,1]^\omega)\) of compact subsets of \([0,1]^\omega\) are the main objects of the investigation. Two such families are equivalent (up to homeomorphism) if each element of one of them has its homeomorphic copy in the other one. Among the elements of the sets \([\mathcal F]\subset\mathcal K\) of families equivalent with \(\mathcal F\), families \(\mathcal B\) of the possibly lowest complexity are examined.
It is recalled that for every analytic family \(\mathcal A\subset\mathcal K\) there is a Polish subspace \(\mathcal P\) of the hyperspace \(\mathcal K\) which is equivalent to \(\mathcal A\), see also [\textit{A. Bartoš} et al., Topology Appl. 266, Article ID 106836, 25 p. (2019; Zbl 1429.54042)]. Using compactifiability studied in the mentioned paper, it is proved that for every \(F_{\sigma}\) family in \(\mathcal K\) there is a closed equivalent family. Some more constructions are needed to get open families \(\mathcal O_n\subset\mathcal K\), \(n\in\omega\), such that the equivalence classes \([\mathcal O_n]\) are distinct and they are the unique classes of equivalence up to homeomorphism which contain an open family.
Corresponding results on families of continua are also derived.
Reviewer: Petr Holický (Praha)The Fell compactification of a poset.https://zbmath.org/1460.540062021-06-15T18:09:00+00:00"Bezhanishvili, G."https://zbmath.org/authors/?q=ai:bezhanishvili.guram"Harding, J."https://zbmath.org/authors/?q=ai:harding.johnThis paper deals with the hit-or-miss topology on the closed sets of a topological space, and the associated Fell compactification of a locally compact \(T_0\)-space, which is the closure of its image in its hit-or-miss topology. The aim is to investigate the hit-or-miss topology, and associated Fell compactification, when it is applied to the locally compact \(T_0\)-space formed by a poset \(P\) with its Alexandroff topology.
The authors show that the closed sets of \(P\) with the hit-or-miss topology form the Priestley space of the bounded distributive lattice freely generated by the order dual of \(P\). The Fell compactification \(H(P)\) is shown to be the Priestley space of a sublattice of the upsets of \(P\). The restriction of the hit topology to \(H(P)\) is a stable compactification of \(P\); when \(P\) is a chain, this is the least stable compactification of \(P\).
For the entire collection see [Zbl 1448.62015].
Reviewer: Jorge Picado (Coimbra)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)