Some generalizations of countable compactness
Let \({\mathcal V}\) be a family of subsets of X. For \(A\subset X\), the star of A relative to \({\mathcal V}]\) denoted by \({\mathcal V}(A)\) is the union of members of \({\mathcal V}\) intersecting A. \textit{H. M. Fleischman} [Fundam. Math. 67, 1-9 (1970; Zbl 0194.546)] called a topological space X starcompact if for every open cover \({\mathcal V}\) of x, there is a finite set \((b_ 1,b_ 2,...,b_ n)\subset X\) such that \(X=\cup^{n}_{i=1}{\mathcal V}(b_ i)\). For \(\alpha <\omega\), the \(\alpha\)-star of A relative to \({\mathcal V}\) [denoted \({\mathcal V}(A;\alpha)]\) is \({\mathcal V}(A;1)={\mathcal A}(A)\) and \({\mathcal V}(A;\alpha +1)={\mathcal V}({\mathcal V}A;\alpha))\) for \(\alpha =1,2,3,... \). Definition: A space x is \(\omega\)-starcompact if for every open cover \({\mathcal V}\) of X there exists a finite family \(\{B_ 1,B_ 2,...,B_ n\}\subset {\mathcal V}\) and an \(\alpha <\omega\) such that \(X=\cup^{n}_{i=1}{\mathcal V}(B;\alpha)\). If the same \(\alpha <\omega\) holds for very open cover of X, then X is called \(\alpha\)-starcompact. The author shows that the class of \(\alpha\)-starcompact and \(\omega\)- starcompact spaces includes the class of countably compact spaces and H- closed spaces. These spaces share several interesting properties of countably compact spaces, including pseudocompactness.
scientific article; zbMATH DE number 4022140
scientific article