\(s\)-expandable spaces (Q1882132)

Let \((X,\tau)\) be a topological space. A subset \(E\) of \(X\) is called to be semi-open if \(B\subset \text{Cl}(\text{Int}(B))\), see \textit{N. Levine} [Am. Math. Mon. 70, 36--41 (1963; Zbl 0113.16304)]. A collection \(\{Fa:a\in I\}\) of subsets of \((X,\tau)\) is said to be \(s\)-locally finite if for each \(x\in X\), there exists a semi-open set \(U\) containing \(x\) and \(U\) intersects \(Fa\) at most for finitely many \(a\). A topological space \((X,\tau)\) is said to be expandable, see \textit{L. L. Krajewski} [Can. J. Math. 23, 58-68 (1971; Zbl 0206.51503)] if for every locally finite collection \(\{Fa:a\in I\}\) of subsets of \(X\) there exists a locally finite collection \(\{Ga:a\in I\}\) of open sets of \(X\) such that \(Fa\subset Ga\) for each \(a\in I\). In this paper the author introduces the notion of \(s\)-expandable space as a variation of expandable spaces. A topological space \((X,\tau)\) is said to be \(s\)-expandable (resp. co-\(s\)-expandable) if for every \(s\)-locally finite collection \(\{Fa:a\in I\}\) (resp. \(| I| \leq \infty\)) of subsets of \(X\) there exists an \(s\)-locally finite collection \(\{Ga:a\in I\}\) of open sets of \(X\), such that \(Fa\subset Ca\) for each \(a\in I\). Every \(s\)-expandable space is expandable. The author proves that a topological space is \(s\)-expandable if every semi-open cover of \(X\) has a locally finite open refinement and that an extremally disconnected semi-regular space is \(s\)-expandable if and only if is expandable. Some properties of \(s\)-expandable spaces are studied.