Continuity Properties of Discontinuous Homomorphisms and Refinements of Group Topologies

Abstract

We present several conditions on topological groups $G$ and $H$ under which \emph{every} discontinuous homomorphism of $G$ to $H$ preserves accumulation points of open sets in $G$. It is also proved that every (locally) precompact abelian group admits a strictly finer zero-dimensional (locally) precompact topological group topology of the same weight as the original one.