## Groups with only resolvable group topologies

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- by W. W. Comfort and Jan van Mill
- Proc. Amer. Math. Soc.
**120**(1994), 687-696 - DOI: https://doi.org/10.1090/S0002-9939-1994-1209097-X
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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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## Bibliographic Information

- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**120**(1994), 687-696 - MSC: Primary 20K45; Secondary 03E35, 03E50, 54G05
- DOI: https://doi.org/10.1090/S0002-9939-1994-1209097-X
- MathSciNet review: 1209097