Hereditary coreflective subcategories of an epireflective subcategory A of Top such that A₂ÍA (here A₂ 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 AÍHaus 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. Činčura: Heredity and coreflective subcategories of the category of topological spaces. Appl. Categ. Structures 9, 131–138 (2001)