Variations on a theorem of Arhangel'skii and Pytkeev
type: Item
instance of: scholarly article
title: Variations on a theorem of Arhangel'skii and Pytkeev
zbMATH Open document ID: 1240.54020
author: Juan-Miguel Gracia
published in: Acta Mathematica Hungarica
publication date: 2011-12-19T00:00:00Z
review text: Arkhangel'skij and Pytkeev proved (for details, see \textit{A. V. Arkhangel'skij} [Topological Function Spaces. Mathematics and its Applications, Soviet Series 78. Dordrecht etc.: Kluwer Academic Publishers (1992; Zbl 0758.46026)]) that the tightness of the space \(C_p(X)\) equals the supremum of the Lindelöf numbers of finite powers of \(X\). If \(\text{Exp}(X)\) denotes the set of all closed subsets of a space \(X\), then the lower Vietoris topology on \(\text{Exp}(X)\) is the one having as a base the family of all subsets of the form \(\{F \in\text{Exp}(X) : F\cap U \neq\emptyset\;\text{for every}\;U\in \mathbb V\}\) where \(\mathbb V\) is a finite set of open subsets of \(X\). The lower Vietoris topology on \(C(X)\), the set of all real functions on an infinite Tychonoff space \(X\) is the one obtained by inheritance from \(\text{Exp}(X \times \mathbb R)\). The point-open topology (or pointwise convergence topology) on \(C(X)\) is the one generated by the family of all sets of the form \(\{f \in C(X): f[M]\subseteq U\}\) where \(M\) ranges over finite subsets of \(X\) and \(U\) over open sets of \(\mathbb R\). For two topologies \(\tau_1\), \(\tau_2\) on \(X\), with \(\tau_1\supseteq\tau_2\), the tightness of \(\tau_1\) relative to \(\tau_2\) is defined to be the least infinite cardinal \(\kappa\) such that whenever \(x\in X\), \(A\subseteq X\), if \(x\in cl_{\tau1}(A)\), then there is a \(B \subseteq A\) of cardinality \(\leq \kappa\) such that \(x\in cl_{\tau2}(B)\). The weak Lindelöf number of a space, \(wl(X)\) is defined as the least infinite cardinal \(\kappa\) such that for each open cover \(V\) of \(X\), there is a \(\mathbb U\subset \mathbb V\) of cardinality \(\leq \kappa\) such that \(cl(\cup\mathbb U) = X\). (A family like this is called a weak cover of \(X\)). Define the following selection principle \(S_{fin}(\mathbb A ,\mathbb B)(S_1(\mathbb A,\mathbb B)\) respectively): for each sequence \(\langle A_n:n\in \mathbb N\rangle\) of elements of \(\mat…
reviewed by: T. Thrivikraman
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54E55
zbMATH DE Number: 5990614
zbMATH Keywords: lower Vietoris topology
zbMATH Keywords: point-open topology
zbMATH Keywords: function space
zbMATH Keywords: weak Lindelöf number
zbMATH Keywords: weak cover
zbMATH Keywords: selection principle
zbMATH Keywords: tightness
zbMATH Keywords: fan tightness
MaRDI profile type: Publication
full work available at URL: s10474-010-0058-y
OpenAlex ID: W1969135896
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DOI: 10.1007/S10474-010-0058-Y
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description: scientific article; zbMATH DE number 5990614