On generalization of \(G_{\delta}\) diagonal and metrization
A cover of a topological space X is called semi-open if \(x\in Int St(x,{\mathbf{U}})\) for \(x\in X\), where St(x,\textbf{U})\(=\cup \{U\in {\mathbf{U}}|\) \(x\in X\}\). A topological space X is said to have \((\alpha_ 1)\) (resp. \((\alpha_ 2))\) if there is a sequence \(\{\) \textbf{U}\({}_ n\}_ n\) of semi-open covers with \(\cap_ nSt(x,{\mathbf{U}}_ n)=\{x\}\) (resp. \(\cup_ n\overline{St(x,{\mathbf{U}}_ n)}=\{x\})\) for each \(x\in X\). This paper studies topological spaces with \((\alpha_ i)\) in connection with the \(G_{\delta}\)-diagonal property defined by \textit{J. Chaber} [Bull. Acad. Pol. Sci., Ser. Sci. Math. 24, 993-998 (1976; Zbl 0347.54013)]. Both \((\alpha_ 2)\) and \(G_{\delta}\) diagonal property imply \((\alpha_ 1)\), but there is a hereditary paracompact space with \((\alpha_ 1)\) which does not have \(G_{\delta}\) diagonal property. An example shows that a \(T_ 2\) compact space satisfying \((\alpha_ 1)\) need not to be metrizable. It is known [cf. \textit{D. Burke}, General Topol. Appl. 2, 287-291 (1972; Zbl 0243.54017)] that \(G_{\delta}\) diagonal property is not preserved under perfect maps. This paper shows that \((\alpha_ 1)\) is preserved under a finite-to-one pseudo-open map. The author then uses this result to show that \(T_ 1\)-semidevelopability is preserved under a continuous finite-to-one pseudo-open maps.
scientific article; zbMATH DE number 4111562
scientific article