Embeddings into (a)-spaces and acc spaces
A space is said to be an (a)-space (or to have property (a)) provided that for every open cover \({\mathcal U}\) of \(X\) and every dense subset \(Y\subset X\) there exists a closed (in \(X\)) discrete subspace \(F\subset Y\) such that \(St(F,{\mathcal U})= X\). This definition was introduced by the author in [Some questions on property (a), Quest. Answers Gen. Topology 15, 103-111 (1997)] and is derived from his notion of absolute countable compactness (acc) [Topology Appl. 58, No. 1, 81-92 (1994; Zbl 0801.54021)] which was derived from a star characterization of countable compactness. Indeed a space is acc if and only if it is a countably compact (a)-space. Like acc, property (a) is not hereditary to closed subsets. This paper shows the failure of closed subsets to inherit property (a) can be dramatic. The main results are (1) every Tychonoff space can be embedded as a closed nowhere dense zero-set in a Tychonoff (a)-space, and (2) every Tychonoff countably compact space can be embedded as a closed nowhere dense zero-set in a Tychonoff absolutely countably compact space.
scientific article; zbMATH DE number 1083979