A note on \(\alpha\)-regular spaces (Q1125480)

Recently, \textit{R, Devi}, \textit{K. Balachandran} and \textit{H. Maki} [ibid. 29, No. 1, 37-49 (1998; Zbl 0927.54018)] have defined a topological space \(X\) to be \(\alpha\)-regular if for any closed subset \(F\) of \(X\) and any point \(x\in X\smallsetminus F\) disjoint \(\alpha\)-open sets \(U\) and \(V\) of \(X\) exist such that \(x\in U\) and \(F\subset V\). They have obtained a characterization and some preservation theorems of \(\alpha\)-regular spaces. The purpose of this note is to show that \(\alpha\)-regularity is equivalent to regularity and to obtain a slight improvement of their preservation theorems.