Borsik's properties of topological spaces and their applications (Q6542430)

Let \(X\) and \(Y\) be topological spaces. We say that a function \(f\colon X\to Y\) is super quasi-continuous if \(f\) restricted to the set of continuity points of \(f\) is dense in \(f\). The family of all such mappings is denoted by \(QC^*(X,Y)\). There are some special properties concerning a topological space \(X\) denoted by \(BP_1\), \(BP_2\) and \(BP_3\). They are described in [\textit{J. Borsík}, Tatra Mt. Math. Publ. 8, 175--184 (1996; Zbl 0914.54008)] and the following facts are also proved in this paper.\N\begin{itemize}\N\item Every metrizable separable space has the property \(BP_1\).\N\item Every pseudometrizable space has the property \(BP_2\).\N\item Every perfectly normal locally connected space has the property \(BP_3\).\N\end{itemize}\NAt the same time, another version of the property \(BP_3\) was introduced by \textit{Z. Grande} [Fundam. Math. 129, No. 3, 167--172 (1988; Zbl 0657.26003)]. We will denote it by \(BP_3'\).\N\NBorsík used the property \(BP_2\) to show that if \(X\) is a pseudo-metrizable topological space then every cliquish function \(f\colon X\to\mathbb{R}\) can be represented as the sum of three quasi-continuous functions. Following this fact, \textit{Ľ. Holá} [Result. Math. 76, No. 3, Paper No. 126, 11 p. (2021; Zbl 1472.54005)] proved recently that if \(X\) is a separable metrizable space in which there is a closed nowhere dense set \(F\subset X\) with the cardinality \(c\), then the family of quasicontinuos non Borel measurable functions from \(X\) to \([0,1]\) has cardinality \(2^c\).\N\NThe present paper is a continuation of Holá's investigation. The author, amongst others, proves the following facts.\N\begin{itemize}\N\item[1.] Let \(X\) be a second countable Hausdorff topological space with the property \(BP_3'\) in which there is a closed nowhere dense set \(F\subset X\) with the cardinality \(c\). Then the family of all \(QC^*\) non Borel measurable functions from \(X\to\{0,1\}\) has cardinality \(2^c\).\N\item[2.] Let \(X\) be a topological space with the property \(BP_3'\) in which there is a closed nowhere dense set \(F\subset X\) which is homeomorphic with the Cantor set. Then\N\N(A) There exists an additive group of size \(c\) contained in the family of non-Borel \(QC^*(X,Z)\) functions plus the zero function;\N\N(B) Let \(K\in \{\mathbb{Q},\mathbb{R},\mathbb{C}\}\). There is a \(c\)-dimensional linear space (over \(K\)) contained in the family of non-Borel \(QC^*(X,K)\) plus the zero function;\N\N(C) There exists a cone contained in the family of non-Borel \(QC^*(X,\mathbb{R})\) plus the zero function that is spaned by \(2^c\) linearly independent generators;\N\N(D) There is a lattice of size \(2^c\) contained in the family of non-Borel \(QC^*(X,\{0,1\})\) functions.\N\end{itemize}\NFinally, the author generalizes the fact (see [\textit{J. Wódka}, Linear Algebra Appl. 459, 454--464 (2014; Zbl 1309.15005)]) that the family of all quasi-continuous functions from \(\mathbb{R}\) to \(\mathbb{R}\) that are not Lebesgue measurable plus the zero function contains a \(c\)-generated free algebra. Precisely, he shows that if \(X\) is a topological space with the property \(BP_3'\) in which there is a closed nowhere dense set \(F\subset X\) which is homeomorphic to the Cantor set, then there exists a \(c\)-generated free algebra (over \(\mathbb{R}\)) contained in the family of non-Borel \(QC^*(X,\mathbb{R})\) functions plus the zero function.