Some applications of discrete selectivity and Banakh property in function spaces
type: Item
instance of: scholarly article
title: Some applications of discrete selectivity and Banakh property in function spaces
zbMATH Open document ID: 1481.54015
published in: European Journal of Mathematics
publication date: 2020-04-15T00:00:00Z
review text: In this work the author deals with, among other things, domination, strong domination, discrete selectivity and inessentiality. The family of all compact subsets of a space \(Z\) is denoted by \(\mathcal{K}(Z)\). A space \(X\) is defined to be dominated by a space \(Y\) if there exists a family \(\mathcal{F} = \{F_K : K \in \mathcal{K}(Y )\}\subset \mathcal{K}(X)\) such that \(\bigcup \mathcal{F} = X\) and \(F_K \subset F_L\) whenever \(K\subset L\). If, additionally, for every compact subspace \(E\subset X\), there exists \(K \in \mathcal{K}(Y)\) such that \(E \subset F_K\), then the space \(X\) is strongly dominated by the space \(Y\). Given a space \(X\), a set \(A\subset X\) is called inessential if either: \(\ast\) there exists a countable set \(B \subset X\setminus A\) such that \((\overline{B}\setminus B)\setminus A \neq \emptyset\), or \(\ast \ast\) there exist countable disjoint subsets \(B\) and \(C\) of the set \(X\setminus A\) such that \((\overline{B}\setminus B) \cap (\overline{C}\setminus C) \neq \emptyset\). The space \(X\) is called essentially uncountable if every countable subset of \(X\) is inessential. Recall that a space \(Z\) is discretely selective if for any sequence \(\{U_n : n \in \omega \}\) of non-empty open subsets of \(Z\), it is possible to choose a point \(x_n\in U_n\) for every \(n \in \omega\) in such a way that the selection \(\{x_n : n \in \omega \}\) is a closed and discrete subset of \(Z\). A space \(Z\) has the Banakh property (see [\textit{M. Krupski} and \textit{W. Marciszewski}, J. Math. Anal. Appl. 452, No. 1, 646--658 (2017; Zbl 1376.54019); \textit{V. V. Tkachuk}, Eur. J. Math. 4, No. 2, 664--675 (2018; Zbl 1406.54010); \textit{V. V. Tkachuk}, Topology Appl. 228, 318--326 (2017; Zbl 1381.54018)]) if there exists a countable family \(\mathcal{F}\) of closed nowhere dense subsets of \(Z\) such that every compact subset of \(Z\) is contained in an element of \(F\). The author proves that an uncountable space \(X\) must be e…
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54C05
Mathematics Subject Classification ID: 46A50
Mathematics Subject Classification ID: 54A25
zbMATH DE Number: 7189612
zbMATH Keywords: domination by a space
zbMATH Keywords: strong domination by a space
zbMATH Keywords: function spaces
zbMATH Keywords: Lindelöf \(\Sigma\)-space
zbMATH Keywords: metrizable space
zbMATH Keywords: Banakh property
zbMATH Keywords: discrete selectivity
zbMATH Keywords: essentially uncountable space
author: Vladimir Tkachuk
reviewed by: Daniel Jardón
MaRDI profile type: Publication
full work available at URL: s40879-019-00342-7
OpenAlex ID: W2944901598
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Wikidata QID: Q127828887
DOI: 10.1007/S40879-019-00342-7
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description: scientific article; zbMATH DE number 7189612