A class of extension properties that are not simply generated
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.
closed-hereditary topological property preserved by products
\({\mathcal P}\)-regular compactification
scientific article; zbMATH DE number 3909308
scientific article