Hereditary Coreflective Subcategories of Categories of Topological Spaces

Abstract

Let $\mathcal{A}$ be an epireflective subcategory of the category Top of topological spaces that is not bireflective (e.g., the category of Hausdorff spaces, the category of Tychonoff spaces) and ℬ be a coreflective subcategory of $\mathcal{A}$ . Extending the corresponding result obtained for coreflective subcategories of Top we prove that ℬ is hereditary if and only if it is closed under the formation of prime factors. As a consequence we obtain that every hereditary coreflective subcategory ℬ of $\mathcal{A}$ containing a non-discrete space is generated by a class of prime spaces and if $\mathcal{A}$ is a quotient-reflective subcategory of Top, then the assignment $\mathcal{B}\mapsto \mathcal{B}\cap \mathcal{A}$ gives a bijection of the collection of all hereditary coreflective subcategories of Top that contain the class FG of all finitely generated spaces onto the collection of all hereditary coreflective subcategories of $\mathcal{A}$ that contain $\mathbf{FG}\cap \mathcal{A}$ . Some applications of these results in the categories of Hausdorff spaces, Tychonoff spaces and zero-dimensional Hausdorff spaces are presented.