Hereditary Coreflective Subcategories of the Categories of Tychonoff and Zero-Dimensional Spaces

Abstract

In this paper the structure of hereditary coreflective subcategories in the categories Tych of Tychonoff and ZD of zero-dimensional spaces is studied. It is shown that there are (many) hereditary additive and divisible subcategories in Tych and ZD which are not coreflective. Moreover, if ${\mathcal{A}}$ is an epireflective subcategory of the category 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. It is also shown, in the case of Tych under Martin’s axiom or under the continuum hypothesis, that if ${\mathcal{B}}$ is a hereditary coreflective subcategory of Tych (ZD), then either the topologies of all spaces belonging to ${\mathcal{B}}$ are closed under countable intersections or it contains all Tychonoff spaces (zero-dimensional spaces) with Ulam nonmeasurable cardinality.