Pseudocompact group topologies with prescribed topological subspaces

Abstract

We prove that every pseudocompact topological Abelian group G admits a pseudocompact topological group topology with a non-trivial convergent sequence. Imposing some restrictions on the properties of G, stronger properties are also obtained. If, for instance, G is an Abelian group with m(β) ≤ r0(G) ≤ |G| ≤ 2β (see the Introduction below for unexplained terminology) for some uncountable cardinal β, and X is any topological space with |X| ≤ r0(G) and w(X) ≤ β, then G admits a pseudocompact topological group topology that contains X as a subspace. If, on the other direction, G is a torsion Abelian group that admits a pseudocompact group topology, then, for every sequence (an)n∈ of G there exists a pseudocompact group topology on G for which some subsequence of (an)n∈ converges.