Selections and approaching points in products. (Q2798072)

For a topological space \(Z\), let \(\mathcal{F}_2(Z)=\{S \subset Z : 1 \leq |S|\leq 2 \}\) with the Vietoris topology. A map \(\sigma : \mathcal{F}_2(Z) \to Z\) is called a weak selection on \(X\) if \(\sigma (\{x,y\}) \in \{x,y\}\) for every \(\{x,y\} \in \mathcal{F}_2(Z)\).NEWLINENEWLINE Let \(X_p\) (resp., \(Y_q\)) denote the space \(X\) (resp., \(Y\)) with only one non-isolated point \(p\in X\) (resp., \(q \in Y\)). Concerning the existence of continuous weak selections on products, \textit{S. García-Ferreira} et al. [Topology Appl. 160, No. 18, 2465--2472 (2013; Zbl 1282.54014)] proved the following two theorems:NEWLINENEWLINE Theorem 1. If \(X_p\times Y_q\) has a continuous weak selection, then \(\psi (q, Y_q) \leq a (p,X_p)\), where \(a(p,X_p)\) is the approaching number of \(p\) in \(X_p\) defined by \(a(p,X_p)= \min \{\kappa : p \in \overline{A} \text{ for some } A \subset X_p \setminus \{p\} \text{ with } |A| \leq \kappa \}\) and \(\psi (q, Y_q)\) is the pseudocharacter of \(q\) in \(Y_q\).NEWLINENEWLINE Theorem 2. If \(S\) is a stationary set in a regular uncountable cardinal and \(a(p, X_p) <|S|\), then \(X_p \times S\) has no continuous weak selection.NEWLINENEWLINE In this paper, the author gives simple self-contained proofs of the above two theorems and another alternative proof of Theorem 1 which is related to a classical result of \textit{M. Katětov} [Fundam. Math. 35, 271--274 (1948; Zbl 0031.28301)] about complete normality of products. The author also proves that if \(X\) is a regular countably compact space such that \((\omega +1) \times X\) has a continuous weak selection, then \(X\) is a compact zero-dimensional first countable space.