Closed discrete selections for sequences of open sets in function spaces
Given a sequence \(\mathcal{U} = \{U_n : n\in\omega\}\) of non-empty open subsets of a topological space \(X\), a set \(\{ x_n : n\in\omega\}\) is a selection of \(\mathcal{U}\) if \(x_n\in U_n\) for every \(n \in\omega\). A space \(X\) is called \textit{discretely selective} if every sequence of non-empty open subsets of \(C_p(X)\) has a closed discrete selection. The author shows that a space \(X\) is uncountable if and only if the space \(C_p(X)\) is discretely selective. The analogous statement is not true for \(C_p(X, [0,1])\) and the main query considered in the paper is when \(C_p(X,\mathbb{I})\) is discretely selective. Given a space \(X\), we will call a set \(A\in X\) \textit{inessential} if either there exists a countably infinite set \(B \in X \setminus A\) such that \((\overline{B} \setminus B) \setminus A\neq \emptyset\), or there exist countably infinite disjoint subsets \(B,C\subset X \setminus A\) such that \((\overline{B} \setminus B) \cap (\overline{C} \setminus C)\neq \emptyset\). A space \(X\) is called \textit{essentially uncountable} if any countable subset of \(X\) is inessential. The author shows that if \(X\) is essentially uncountable, then \(C_p(X, \mathbb{I})\) is discretely selective. In the main theorem of the paper he proves that every uncountable Lindelöf \(\Sigma\)-space is essentially uncountable, and consequently the space \(C_p(X,\mathbb{I})\) is discretely selective. The discrete selectivity of the space \(C_p(X, \mathbb{I})\) does not imply in general that \(X\) is essentially uncountable. However, for \(\omega\)-monolithic spaces of countable tightness essential uncountability of \(X\) turns out to be equivalent to discrete selectivity of \(C_p(X,\mathbb{I})\). In particular, a metrizable space \(X\) is essentially uncountable if and only if \(C_p(X, \mathbb{I})\) is discretely selective. The paper ends with 11 open problems related to this subject.
scientific article; zbMATH DE number 6850202