Embeddings into (a)-spaces and acc spaces
type: Item
instance of: scholarly article
title: Embeddings into (a)-spaces and acc spaces
zbMATH Open document ID: 0891.54008
DOI: 10.1016/S0166-8641(96)00160-5
published in: Topology and its Applications
publication date: 1998-07-19T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 54C25
Mathematics Subject Classification ID: 54D30
zbMATH DE Number: 1083979
author: Mikhail Matveev
reviewed by: Jerry E. Vaughan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1083979