Hereditary coreflective subcategories of the categories of Tychonoff and zero-dimensional spaces
The author investigates hereditary coreflective subcategories in the categories \textbf{Tych} of Tychonoff and \textbf{ZD} of zero-dimensional spaces. It is shown that there are numerous hereditary additive and divisible subcategories in \textbf{Tych} and \textbf{ZD} that are not coreflective. If a hereditary coreflective subcategory \({\mathcal B}\) of \textbf{Tych} (\textbf{ZD}) contains all countable Tychonoff (zero-dimensional) spaces, then \({\mathcal B}\) contains all Tychonoff (zero-dimensional) spaces of nonmeasurable cardinality (in the sense of Ulam). It is also proved, in the case of \textbf{Tych} under Martin's axiom or under the continuum hypothesis, that if \({\mathcal B}\) is a hereditary coreflective subcategory of \textbf{Tych} (\textbf{ZD}), then either the topologies of all spaces belonging to \({\mathcal B}\) are closed under countable intersections or \({\mathcal B}\) contains all Tychonoff (zero-dimensional) spaces of nonmeasurable cardinality. Finally, it is established that if \({\mathcal A}\) is an epireflective subcategory of the category \textbf{Top} of topological spaces which is not bireflective and \({\mathcal B}\) is an additive and divisible subcategory of \({\mathcal A}\) which is not coreflective, then the coreflective hull of \({\mathcal B}\) in \({\mathcal A}\) is not hereditary.
hereditary additive and divisible subcategory
scientific article; zbMATH DE number 6252057
scientific article