On spaces with a quasi-\(G_\delta(2)\)-diagonal
A sequence \(({\mathcal U}_n)_{n <\omega}\) of open collections of a \(T_1\)-space \(X\) is called a quasi-\(G_\delta (2)\)-diagonal if for every pair \(x,y\) of distinct points in \(X\) there is some \(n<\omega\) such that \(\text{St} (x,{\mathcal U}_n)\neq \emptyset\) and \(y\not\in \text{St(St} (x,{\mathcal U}_n),{\mathcal U}_n)\). It is shown that every \(T_1\)-space with a quasi-\(G_\delta(2)\)-diagonal is a \(c\)-semi-stratifiable space in the sense of \textit{H. W. Martin} [Can. J. Math. 25, 840-841 (1973; Zbl 0247.54031)]. A \(T_1\)-space is a Nagata space if and only if it is a \(q\), quasi-\(N\)-space with a quasi-\(G_\delta(2)\)-diagonal. This result generalizes a recent theorem of \textit{A. M. Mohamad} [Topol. Proc. 24, 215-232 (1999; Zbl 0962.54024)]. A \(T_1\)-space \(X\) with a quasi-\(G_\delta(2)\)-diagonal is semi-stratifiable if and only if it is a \(\beta\)-space. It is shown that from this theorem three recent results of A. M. Mohamad can be deduced.
scientific article; zbMATH DE number 1625866