Hereditary, additive and divisible classes in epireflective subcategories of Top

TL;DR

This paper investigates the properties of hereditary, additive, and divisible classes within epireflective subcategories of Top, extending previous results to broader classes closed under sums and quotients, with applications to the lattice of subcategories.

Contribution

It generalizes the characterization of hereditary subcategories to those closed under sums and quotients, providing new methods and insights into the structure of epireflective subcategories of Top.

Findings

Hereditary subcategories are characterized by closure under prime factors.

The main claim holds under certain conditions, such as containing specific types of spaces.

New results on the lattice structure of subcategories of Top and ZD.

Abstract

Hereditary coreflective subcategories of an epireflective subcategory A of Top such that I_2\notin A (here I_2 is the 2-point indiscrete space) were studied in [C]. It was shown that a coreflective subcategory B of A is hereditary (closed under the formation of subspaces) if and only if it is closed under the formation of prime factors. The main problem studied in this paper is the question whether this claim remains true if we study the (more general) subcategories of A which are closed under topological sums and quotients in A instead of the coreflective subcategories of A. We show that this is true if A \subseteq Haus or under some reasonable conditions on B. E.g., this holds if B contains either a prime space, or a space which is not locally connected, or a totally disconnected space or a non-discrete Hausdorff space. We touch also other questions related to such subclasses of A. We introduce a method extending the results from the case of non-bireflective subcategories (which was studied in [C]) to arbitrary epireflective subcategories of Top. We also prove some new facts about the lattice of coreflective subcategories of Top and ZD. [C] J. \v{C}in\v{c}ura: Heredity and coreflective subcategories of the category of topological spaces. Appl. Categ. Structures 9, 131-138 (2001)