Some results on property (a) (Q1962113)

A space \(X\) is said to have property (a) [\textit{M. V. Matveev}, Some questions on property (a), Quest. Answers Gen. Topology 15, No. 2, 103-111 (1997)] if whenever \(\mathcal U\) is an open cover and \(D\) a dense set one can find a closed discrete set \(F\subseteq D\) such that \(\text{St}(F,\mathcal U)=X\). Property (wa) is the same, but without the word `closed'. The authors present a number of examples that answer questions posed by Matveev [loc. cit.]. (1) There is a normal space without property (a) can be constructed from a Q-set; (2) there is a Tychonoff space without property (wa); (3) if \(X\subseteq\mathbb R\) then the Pixley-Roy subspace \(F_{\leq 2}[X]\) has property (a) but the full Pixley-Roy space \(F[X]\) does not, whenever \(X\) has cardinality \(2^{\aleph_0}\); (4) \(\text{MA}_{\aleph_1}\) implies that Aronszajn trees have property (a) and \(\lozenge\) implies that special Aronszajn trees do not. The paper closes with some remarks about (a)-Dowker spaces: spaces with property (a) whose product with a converging sequence does not have property (a).