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Recently, I have been "updating" myself in the field of topological groups, and, in doing this, I remembered some questions I had a few years ago that I never solved.

First, is there any application of group theory in the study of finite topological spaces? For example, by a group action, one may look at some "symmetries" of a finite topological space, perhaps applying this to the problem of counting the topologies on a finite set.

Second, is there any interesting inverse limit of some finite (non-trivial) topological groups. By interesting I mean that appears in practice. By non-trivial I mean not to end constructing the $p$-adics or Zee-hat.

I looked in a few books, and I did a quick search in mathscinet, and found nothing.

Sorry if this are well-known results. I'm not an expert in this fields.

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    $\begingroup$ You may know that any finite CW complex has the weak homotopy type of a finite topological space. I wonder if a similar statement holds for finite $G$-CW complexes? $\endgroup$
    – Mark Grant
    Commented May 23, 2017 at 16:34
  • $\begingroup$ The main theorem of this paper ("Finite Homogeneous Spaces" by Fora and Al-Bsoul, 1997) shows that finite topological spaces on which homeomorphisms act transitively are not very interesting. So I think we can say that "interesting" finite topological spaces can't have too many symmetries (self-homeomorphisms). $\endgroup$
    – Gro-Tsen
    Commented May 23, 2017 at 17:11
  • $\begingroup$ Thank you very much. Perhaps the problem of counting topologies on a finite set is "just" a combinatorial one. $\endgroup$
    – efs
    Commented May 24, 2017 at 13:28

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