\(S\)-sets and co-\(S\)-closed topologies
A topological space \((X,\tau)\) is defined to be \(S\)-closed if every cover of \(X\) by semi-open sets contains a finite subfamily the closures of whose members cover \(X\). Recall that a subset \(B\) of \((X,\tau)\) is semi- open if \(B\subset\text{cl}(\text{int}B)\). A subset \(S\) of \((X,\tau)\) is defined to be an \(S\)-set in \((X,\tau)\) if every cover of \(S\) by regular closed sets of \((X,\tau)\) contains a finite subcover for \(S\). For any space \((X,\tau)\) the family \(\{U: U\in\tau\) and \(X-U\) is an \(S\)-set in \((X,\tau)\}\) is a base for a coarser topology \(\tau^*\) on \(X\), which is called the co-\(S\)-closed topology of \((X,\tau)\). One of the main results of this paper shows that a space \((X,\tau)\) is \(S\)-closed if and only if \(\tau_ s\), the semi-regularization topology of \(\tau\), satisfies \(\tau_ s\subset\tau^*\). A space \((X,\tau)\) is defined to be subweakly-\(T_ 2\) if \(\tau\text{ cl}\{x\}=\tau_ s\text{cl}\{x\}\) for every point \(x\in X\). The authors prove that \((X,\tau)\) is subweakly-\(T_ 2\) if and only if \(\tau^*\subset\tau_ s\). \textit{A. H. Stone} [Am. J. Math. 82, 900-916 (1960; Zbl 0099.173) and ibid. 86, 888 (1964)] defined a space to be semi-irreducible if every family of nonempty pairwise disjoint open sets is finite. This paper provides the following interesting characterization of such spaces: A space \((X,\tau)\) is semi-irreducible if and only if every subset of \(X\) is an \(S\)-set.
scientific article; zbMATH DE number 97817
scientific article