Almost p-regular spaces and some functions
Some modifications of open and closed sets of a topological space are used for definition and investigation of spaces generalizing regular spaces as almost-regular, p-regular and almost p-regular spaces generalizing both of them introduced by \textit{S. R. Malghan} and \textit{G. B. Navalagi} [Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 34(82), No. 4, 317-326 (1990; Zbl 0752.54005)]. Using for a subset \(A\) of a space \(X\) the notations \(\text{Cl}\) and Int for the closure and the interior, respectively, we call \(A\) regular open (resp. regular closed, preopen, preclosed) if \(A=\text{Int}(\text{Cl}(A))\) (resp. \(A=\text{Cl}(\text{Int}(A))\), \(A\subseteq \text{Int}(\text{Cl}(A))\), complement of \(A\) is preopen). The space \(X\) is said to be almost p-regular if for each regular closed set \(A\) of \(X\) and each point \(x\in X-A\) there exist preopen sets \(U\), \(V\) of \(X\) such that \(x\in U\), \(A\subseteq V\), and \(U\cap V=\emptyset\). Let \(f\) be a map from a space \(X\) in a space \(Y\). The map \(f\) is called {(1)} R-map if \(f^{-1}(V)\) is regular open for every regular open \(V\) of \(Y\), {(2)} M-preopen if \(f(U)\) is preopen for every preopen \(U\) of \(X\), {(3)} pre rgp-closed if \(\text{pCl} f(A)\subseteq U\) for each regular open set \(U\) of \(Y\) with \(f(A)\subseteq U\), where \(A\) is an arbitrary preclosed set of \(X\) and \(\text{pCl} f(A)\) is the intersection of all preclosed sets of \(Y\) containing \(f(A)\). The core of this article is an investigation of the theorems on preservation of almost p-regularity under special maps. The main result is the following theorem (Theorem 5.5): ``Let \(f\) be an M-preopen and pre rgp-closed R-map from a space \(X\) onto a space \(Y\). If \(X\) is almost p-regular, then \(Y\) is almost p-regular''.
scientific article; zbMATH DE number 1244261