Covering properties which, under weak diamond principles, constrain the extents of separable spaces (Q625996)

A topological space is an (a)-space if, for every open cover \(\mathcal U\) of \(X\) and every dense subset \(F\) of \(D\), there is a closed and discrete subset \(F\) of \(D\) such that \(\text{St}(F,\mathcal U)=X\). The authors show that the principle \( \lozenge (\omega,<)\) implies that there is no separable, \(T_1\), locally countably compact (a)-space with uncountable closed, discrete subsets, and that the principle \(\Phi(\omega,<)\) implies that there is no separable, \(T_1\), countably paracompact space with uncountable extent.