Absolutely countably compact spaces (Q1330406)

This paper introduces an interesting new star covering property. A space \(X\) is called absolutely countably compact (acc) provided for every open cover \({\mathcal U}\) of \(X\) and every dense \(Y\subset X\), there exists a finite set \(F\subset Y\) such that \(\bigcup \{U\in {\mathcal U}\): \(U\cap F \neq\emptyset\} =X\). Every compact space is acc, and every acc \(T_ 2\)- space is countably compact. It is, therefore, somewhat unexpected that the acc property is not necessarily preserved by continuous mappings, products with compact spaces, or passage to closed subsets (indeed every countably compact space can be embedded as a closed subset of some acc space). Every countably compact space of countable tightness is acc, as is every \(\Sigma\)-product of compact spaces of countable tightness. If \(X\) is a \(T_ 1\) non-compact space, then there exists a cardinal \(\tau\) such that the product \(X^ \tau\) is not acc. A space \(X\) is called hacc if every closed subset of \(X\) is acc. The product of a \(T_ 2\) hacc (acc) space with a compact first countable space is hacc (acc). Several other related star covering properties are briefly considered.