All topological spaces under consideration are \(T_1\)-spaces. If \({\mathcal K}\) is a cover of a space \(X\), then a cover \({\mathcal P}\) is called a (mod \({\mathcal K}\))-network for \(X\) if, whenever \(K\subseteq U\) with \(K\in {\mathcal K}\) and \(U\) open in \(X\), then \(K\subseteq P\subseteq U\) for some \(P\in {\mathcal P}\). A space \(X\) is a \(\Sigma\)-space (\(\Sigma^*\)-space) if it has a \(\sigma\)-locally finite \((\sigma\)-hereditarily closure-preserving) closed (mod \({\mathcal K}\))-network for some cover \({\mathcal K}\) of \(X\) by countably compact sets. If in the definitions \({\mathcal K}\) is a cover of \(X\) consisting of compact sets, then \(X\) is called a strong \(\Sigma\)-space (strong \(\Sigma^*\)-space).
In this article some conditions are discussed that imply the \(\Sigma^*\)-property. Answering a question of Lin, the author also shows that each strong \(\Sigma^*\)-space in which every point is a \(G_\delta\)-set is a strong \(\Sigma\)-space and concludes that a Hausdorff space is a \(\sigma\)-space if and only if it is a \(\Sigma^*\)-space with a \(G_\delta\)-diagonal.