Some applications of discrete selectivity and Banakh property in function spaces
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 essentially uncountable whenever its extent and tightness are countable. As a consequence, the equality \(\operatorname{ext}(X) = t(X) = \omega\) implies that the space \(C_p(X, [0, 1])\) is discretely selective. It is also proved that if \(X\) is a metrizable space, then \(C_p(X, [0, 1])\) has the Banakh property if and only if so does \(C_p(Y,[0,1])\) for some closed separable \(Y\subset X\). The author applies these results to show that, for a metrizable \(X\), the space \(C_p(X,[0, 1])\) is strongly dominated by a second countable space if and only if \(X\) is homeomorphic to \(D \oplus M\) where \(D\) is a discrete space and \(M\) is countable. For a metrizable space \(X\), the author also proves that \(C_p(X,[0, 1])\) has the Lindelöf property if and only if the set of non-isolated points of \(X\) is second countable. The results of this work solve several open questions from [\textit{V. V. Tkachuk}, Acta Math. Hung. 154, No. 1, 56--68 (2018; Zbl 1399.54047)] in a stronger form. The paper ends with some interesting open problems.
scientific article; zbMATH DE number 7189612
scientific article