On countably ${\mu}$-paracompact spaces

Abstract

A topological space X is countably paracompact if and only if X satisfies the condition (A): For any decreasing sequence {F _i } of non-empty closed sets with ${\bigcap_{i=1}^{\infty} F_{i} = \emptyset}$ there exists a sequence {G _i } of open sets such that ${\bigcap_{i=1}^{\infty}\overline{G_{i}}=\emptyset}$ and ${F_{i} \subset G_{i}}$ for every i. We will show, by an example, that this is not true in generalized topological spaces. In fact there is a ${\mu}$-normal generalized topological space satisfying the analogue of A which is not even countably ${\mu}$-metacompact. Then we study the relationships between countably ${\mu}$-paracompactness, countably ${\mu}$-metacompactness and the condition corresponding to condition A in generalized topological spaces.