Selection topologies
For a topological space \(X\), let \(\mathcal{F}_2(X)\) denote the set of all non-empty subsets of \(X\) with at most two points. A map \(\sigma : \mathcal{F}_2(X) \to X\) is called a weak selection on \(X\) if \(\sigma(F) \in F\) for each \(F \in \mathcal{F}_2(X)\). A weak selection on \(X\) is said to be continuous if it is continuous with respect to the Vietoris topology on \(\mathcal{F}_2(X)\). For a weak selection \(\sigma\) on \(X\) and \(x \in X\), put \((\leftarrow , x)_\sigma = \{ y \in X : \sigma(\{x,y \}) =y \text{ and } x \neq y \}\) and \((x, \rightarrow )_\sigma = \{y \in X : \sigma(\{x,y \}) =x \text{ and } x \neq y \}\). The topology \(\tau_\sigma\) induced by the subbase consisting of all intervals \((\leftarrow , x)_\sigma\) and \((x, \rightarrow )_\sigma\), \(x \in X\), is called the selection topology induced by \(\sigma\), which was introduced by the author and \textit{T. Nogura} [Appl. Gen. Topol. 2, No. 2, 205--218 (2001; Zbl 0993.54019)]. A weak selection \(\sigma\) on a space \((X, \tau)\) is said to be separately continuous if \(\tau_\sigma \subset \tau\). Every continuous weak selection is separately continuous. In [\textit{S. García-Ferreira} et al., Houston J. Math. 39, No. 4, 1385--1399 (2013; Zbl 1287.54003)], S. García-Ferreira, K. Miyazaki, T. Nogura and A. H. Tomita introduced the notions of a weak selection space and a continuous weak selection space as generalizations of a topological space with a selection topology. A space \((X,\tau)\) is called a weak selection space (resp., continuous weak selection space) if there is a set \(\Sigma\) of weak selections (resp., continuous weak selections) on \(X\) such that \(\tau\) is equal to the smallest topology on \(X\) containing \(\tau_\sigma\) for all \(\sigma \in \Sigma\). In this paper, the author prove that a space with a separately continuous (resp., continuous) weak selection is a weak selection space (resp., continuous weak selection space) if and only if each of its components is orderable and has a clopen neighborhood base. Here, a space \((X,\tau)\) is said to be orderable if \(\tau\) is the order topology generated by some linear order on \(X\), and a subset \(C\) of a space \(X\) is said to have a clopen neighborhood base if for each neighborhood \(U\) of \(C\) there exists a clopen set \(H\) of \(X\) such that \(C\subset H \subset X\). A space is said to be suborderable if it is a subspace of some orderable space. By applying the above theorem and \textit{S. Purisch}'s characterization of suborderability of metrizable spaces [Trans. Am. Math. Soc. 226, 59--76 (1977; Zbl 0308.54023)], the author proves that for a second countable space \(X\) the following three conditions are equivalent: (a) \(X\) is a weak selection space; (b) \(X\) is suborderable; (c) \(X\) is a continuous weak selection space. A space \((X,\tau)\) is a said to be weakly orderable if \(\tau\) contains the order topology generated by some linear order on \(X\). Every suborderable space is weakly orderable, and every weakly orderable space has a continuous (and hence, separately continuous) weak selection. The author also proves that every second countable space with a separately continuous weak selection is weakly orderable, which extends a theorem in [\textit{V. Gutev}, Fundam. Math. 196, No. 3, 275--287 (2007; Zbl 1129.54016)].
scientific article; zbMATH DE number 6522004