A class of extension properties that are not simply generated
type: Item
instance of: scholarly article
title: A class of extension properties that are not simply generated
zbMATH Open document ID: 0569.54025
DOI: 10.1016/0166-8641(85)90017-3
author: R. G. Woods
published in: Topology and its Applications
publication date: 1985-01-01T00:00:00Z
review text: Let \({\mathcal P}\) be a closed-hereditary topological property preserved by products. Call a space \({\mathcal P}\)-regular if it is homeomorphic to a subspace of a product of spaces with \({\mathcal P}\). Suppose that each \({\mathcal P}\)-regular space possesses a \({\mathcal P}\)-regular compactification. It is well-known that each \({\mathcal P}\)-regular space X is densely embedded in a unique space \(\gamma_{{\mathcal P}}X\) with \({\mathcal P}\) such that if \(f: X\to Y\) is continuous and Y has \({\mathcal P}\), then f extends continuously to \(\gamma_{{\mathcal P}}X\). Call X \({\mathcal P}\)-pseudocompact if \(\gamma_{{\mathcal P}}X\) is compact. Associated with \({\mathcal P}\) is another topological property \({\mathcal P}^{\#}\), possessing all the properties hypothesized for \({\mathcal P}\) above, defined as follows: a \({\mathcal P}\)- regular space X has \({\mathcal P}^{\#}\) if each \({\mathcal P}\)-pseudocompact closed subspace of X is compact. It is known that the \({\mathcal P}\)- pseudocompact spaces coincide with the \({\mathcal P}^{\#}\)-pseudocompact spaces, and that \({\mathcal P}^{\#}\) is the largest closed-hereditary, productive property for which this is the case. We prove that if \({\mathcal P}\) is not the property of being compact and \({\mathcal P}\)-regular, then \({\mathcal P}^{\#}\) is not simply generated; in other words, there does not exist a space E such that the spaces with \({\mathcal P}^{\#}\) are precisely those spaces homeomorphic to closed subspaces of powers of E.
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54C20
zbMATH DE Number: 3909308
zbMATH Keywords: E-compactness
zbMATH Keywords: extension property
zbMATH Keywords: closed-hereditary topological property preserved by products
zbMATH Keywords: \({\mathcal P}\)-regular space
zbMATH Keywords: \({\mathcal P}\)-regular compactification
zbMATH Keywords: \({\mathcal P}\)-pseudocompact spaces
MaRDI profile type: Publication
full work available at URL: 0166-8641(85)90017-3
OpenAlex ID: W1968286809
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description: scientific article; zbMATH DE number 3909308