\(D\)-spaces and thick covers
This paper gives over 15 results relating nearly a dozen subtly defined properties. For readability, this review mentions only a few of these results relating only a few of these properties. The results mentioned in this review all end in ``then \(X\) is a \(D\)-space,'' but many of the results in this paper do not have this format. Recall that \(X\) is a \(D\)-space iff, given an open cover \(\{U_x: x \in X\}\) where each \(x \in U_x\), there is a closed discrete \(D \subseteq X\) so that \(X = \bigcup_{x \in D}U_x\). To give the flavor of this paper, here are three results: \textit{If \(X\) has an almost thick cover by closed Lindelöf \(D\)-sets, then \(X\) is a \(D\)-space.} Here a cover \(\mathcal L\) is almost thick iff \(\forall H \in [X]^{< \omega}\) there is \(L_H\) a finite union of elements of \(\mathcal L\) so that if \(A \subset X\) and \(A\) is not closed then there is \(H \in [A]^{< \omega}\) with \(L_H \cap\) cl\((A \setminus A) \neq \emptyset\). \textit{If \(X\) is \(t\)-metrizable, then \(X\) is a \(D\)-space.} Here a space \((X, \tau)\) is \(t\)-metrizable iff it has a finer metrizable topology \(\pi\) and a function \(J: [X]^{< \omega} \rightarrow [X]^{< \omega}\) so that if \(A \subset X\) then cl\(_{\tau}A \subset\) cl\(_{\pi}\bigcup_{H \in [A]^{< \omega}}J(H)\). \textit{If \(X\) is a union of finitely many screenable \(\sigma\)-spaces, then \(X\) is a \(D\)-space.} Here a \(\sigma\)-space is one with a \(\sigma\)-discrete closed network; a space is screenable iff each open cover has a \(\sigma\)-pairwise disjoint open refinement.
scientific article; zbMATH DE number 5956057
scientific article