There are \(2^{\mathfrak{c}}\) quasicontinuous non Borel functions on uncountable Polish space
type: Item
instance of: scholarly article
title: There are \(2^{\mathfrak{c}}\) quasicontinuous non Borel functions on uncountable Polish space
zbMATH Open document ID: 1472.54005
published in: Results in Mathematics
publication date: 2021-07-16T00:00:00Z
review text: A function \(f : X \to Y\) between spaces \(X\) and \(Y\) is said to be quasicontinuous if \(f^{-1}(V) \subseteq \mathrm{cl}_X(\mathrm{int}_X(f^{-1}(V)))\) for every open \(V \subseteq Y\) where \(\mathrm{cl}_X\) and \(\mathrm{int}_X\) represent the closure and interior operators in \(X\), respectively. \textit{S. Marcus} proved [Colloq. Math. 8, 47--53 (1961; Zbl 0099.04501)], that there is a quasicontinuous function from the interval \([0,1]\) to \(\mathbb R\) which is not Lebesgue measurable. In [\textit{Ľ. Holá}, Am. Math. Mon. 128, No. 5, 457--460 (2021; Zbl 1471.28001)], it was shown that there are \(2^{\mathfrak c}\) such functions. In this paper, using embeddings of the irrationals or the interval \([0,1]\), it is shown that, if \(X\) is either an uncountable Polish space or a locally pathwise connected perfectly normal space with at least one non-isolated point, then there are \(2^{\mathfrak c}\) quasicontinuous non-Borel functions from \(X\) to \([0,1]\). In the remainder of the paper, properties of cliquish functions and set-valued maps are investigated. For a space \(X\) and a metric space \((Y,d)\), a function \(f: X \to Y\) is cliquish if, for any \(x \in X\), every \(\varepsilon > 0\), and every neighborhood \(U\) of \(x\), there is a nonempty open set \(G \subseteq U\) so that \(d(f(x_1),f(x_2)) < \varepsilon\) for every \(x_1,x_2 \in G\). Though quasicontinuous functions into a metric space are cliquish, not all cliquish functions are quasicontinuous. In this paper it is established that, if \(X\) is a space in which every open set is an \(F_\sigma\), then \(X\) is a Baire space if and only if every Borel measurable function of the first class from \(X\) to \(\mathbb R\) is cliquish. The final results of this paper establish new characterizations of minimal usco set-valued mappings in metric spaces by studying quasicontinuous Borel measurable functions.
reviewed by: Christopher Caruvana
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 54C08
Mathematics Subject Classification ID: 54C60
zbMATH DE Number: 7372264
zbMATH Keywords: quasicontinuous function
zbMATH Keywords: Borel measurable function
zbMATH Keywords: cliquish function
zbMATH Keywords: Polish space
zbMATH Keywords: minimal set-valued mapping
zbMATH Keywords: selection
author: Ľubica Holá
MaRDI profile type: Publication
full work available at URL: s00025-021-01440-3
OpenAlex ID: W3169680925
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DOI: 10.1007/S00025-021-01440-3
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description: scientific article; zbMATH DE number 7372264