There are \(2^{\mathfrak{c}}\) quasicontinuous non Borel functions on uncountable Polish space
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.
scientific article; zbMATH DE number 7372264
scientific article