Follow up to this follow up question. Is there an indecomposable group $G$ ie, a group that can't be written in the form $A\times B$ where both $A$ and $B$ are nontrivial groups along with a topology on it that makes it homeomorphic to the Baire space $\mathbb N ^{\mathbb N}$ and makes it into a topological group?
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Why Close vote to this question? If anything, i think my previous questions of which this is a follow up would be the ones that could be closed as they turned out to be trivial. Hopefully this one is interesting unlike those.. – Carla only proves trivial prop Aug 05 '22 at 17:36
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Look, there's really just a lot of ways to modify the construction such that the result we get is still homeomorphic to the Baire space. We haven't used semidirect products yet! – Qiaochu Yuan Aug 05 '22 at 17:43
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That answers my question. Should i delete it? – Carla only proves trivial prop Aug 05 '22 at 17:44
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Well, again, there's still work to do to prove that such a thing is actually indecomposable. I haven't thought through any specific example in detail. – Qiaochu Yuan Aug 05 '22 at 17:48
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Isn't $S_\infty$ with its usual topology (pointwise convergence) homeomorphic to the Baire space? – Alessandro Codenotti Aug 07 '22 at 08:36