Countable Connected Hausdorff groups

Question

Does there exist a countable connected Hausdorff (nontrivial) topological group?

I'm aware that countable connected Hausdorff spaces exist (for example the Bing space, the Golomb space, or $\mathbb{Q}P^\infty$), but can any such spaces be made into a topological group?

I also wonder, if such groups do exist, whether any of the groups $\mathbb{Z}^n$, free groups on at most countably many generators, or $\mathbb{Q}$ be given a compatible connected Hausdorff topology?

Answer 1

No.

Breaking that down: every connected $T_3$ space besides the singleton is uncountable. All Hausdroff topological groups are $T_3$.