Groups with only resolvable group topologies

Abstract:

Adapting terminology suggested by work of E. Hewitt [Duke Math. J. 10 (1943), 309-333], we say that a group $G$ is strongly resolvable if for every nondiscrete Hausdorff group topology $\mathcal {I}$ on $G$ there is $D \subseteq G$ such that both $D$ and $G\backslash D$ are $\mathcal {I}$-dense in $G$. Theorem. Let $G$ be an Abelian group. (a) If $G$ contains no subgroup isomorphic to the group ${ \bigoplus _\omega }\{ 0.1\}$, then $G$ is strongly resolvable. (b) Assume MA. If $G$ contains a copy of ${ \bigoplus _\omega }\{ 0,1\}$, then $G$ is not strongly resolvable. Our proof of (b) depends heavily on work of Malykhin.