On properties of spaces defined in terms of semi-regular sets (Q1964626)

A subset \(A\) of a topological space \(X\) is called semi-\(\theta\)-closed in \(X\) if sCl\(_\theta(A)=A\), where sCl\(_\theta(A)\) is the semi-\(\theta\)-closure of \(A\) defined by \textit{G. Di Maio} and \textit{T. Noiri} [Indian J. Pure Appl. Math. 18, 226-233 (1987; Zbl 0625.54031)]. In this paper, the authors define a generalized semi-\(\theta\)-closed set \(A\) of a space \(X\) to be a subset \(A\) such that sCl\(_\theta(A)\subseteq U\) holds for every open set in \(X\) with \(A\subseteq U\). A generalized semi-\(\theta\)-open set is the complement of a generalized semi-\(\theta\)-closed set. Using these notions, they prove the following statements: (1) A space \(X\) is semi-Hausdorff if and only if every generalized semi-\(\theta\)-closed set of \(X\) is semi-\(\theta\)-closed; (2) a space \(X\) is \(s\)-regular if and only if \(\text{sCl}_\theta(A)\subseteq \text{Cl}(A)\) for every subset \(A\) of \(X\); and (3) a space \(X\) is \(s\)-normal if and only if for every closed set \(F\) and for every open set \(V\) containing \(F\), there exists a generalized semi-\(\theta\)-open set \(U\) of \(X\) such that \(F\subseteq U\subseteq \text{sCl}_\theta(U)\subseteq V\). Preservation of \(s\)-regular spaces and \(s\)-normal spaces under certain maps is also investigated.