Continuous homomorphic images to point-separating classes of topological groups

Abstract

A topological group G is said to be a (C→D)-group if each continuous homomorphic image of G to an arbitrary group contained in the class C is automatically contained in D as well. When C is the class of compact groups, and D is the singleton class of the trivial group, the corresponding class of (C→D)-groups coincides with the traditional minimally almost periodic (MinAP) groups of von Neumann. In this paper we characterize classes C of topological groups for which the corresponding (C→D)-groups can be described in the following way when D is closed under continuous homomorphisms: G is a (C→D)-group if and only if all topological group quotients of G contained in C are automatically contained in D. As an application, we characterize the algebraic structure of Abelian groups which admit a group topology where all continuous homomorphic images to totally disconnected groups are all algebraically torsion groups, or bounded groups.