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Groups with only resolvable group topologies

Authors: W. W. Comfort and Jan van Mill
Journal: Proc. Amer. Math. Soc. 120 (1994), 687-696
MSC: Primary 20K45; Secondary 03E35, 03E50, 54G05
MathSciNet review: 1209097
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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.

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Keywords: Dense subset, resolvable topological space, strongly resolvable group, Boolean group, extremally disconnected space, extremally disconnected topological group, maximal topology, Abelian group, divisible hull
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