Butterfly points and hyperspace selections (Q6640035)

Given an infinite Hausdorff topological space \(X\), let \({\mathcal F}(X)\) be the set of all nonempty closed subsets of \(X\). We endow \({\mathcal F}(X)\) with the Vietoris topology and call it the Vietoris hyperspace of \(X\). Let us recall that a map \(f:{\mathcal F}(X) \to X\) is a selection for \({\mathcal F}(X)\) if \(f(S) \in S\) for every \(S \in {\mathcal F}(X)\), and \(f\) is called continuous if it is continuous with respect to the Vietoris topology on \({\mathcal F}(X)\). Furthermore, a point \(p \in X\) is called countably-approachable if it is either isolated or there exists an open set \(U \subseteq X \setminus \{p\}\) such that \(\overline{U} = U \cup \{p\}\) and \(p\) has a countable clopen base in \(\overline{U}\).\N\NIn an earlier paper [in: Recent progress in general topology III. Based on the presentations at the Prague symposium, Prague, Czech Republic, 2001. Amsterdam: Atlantis Press. 535--579 (2014; Zbl 1320.54002)], the author posed the following open question: Let \(X\) be a space which has a clopen \(\pi\)-base. Let \(p \in X\) be the limit of some sequence \(\{x_n: n \ge 1\} \subseteq X \setminus \{p\}\) and \(p = f(X)\) for some continuous selection \(f\) for \({\mathcal F}(X)\). Then, is it true that \(p\) must be countably approachable? In this paper, the author first provides a purely topological description of the points \(p\in X\) with the property that \(p = f(X)\) for some continuous selections \(f\) for \({\mathcal F}(X)\). As a consequence of this work, the author is able to give a complete affirmative solution to this open question.