On countably μ -paracompact spaces

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 ⋂i=1∞Fi=∅ there exists a sequence {G_i} of open sets such that ⋂i=1∞Gi¯=∅ and F_i⊂ 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 μ-normal generalized topological space satisfying the analogue of A which is not even countably μ-metacompact. Then we study the relationships between countably μ-paracompactness, countably μ-metacompactness and the condition corresponding to condition A in generalized topological spaces.