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I already prove that if $X$ is Hausdorff and compact space $\operatorname{Homeo}(X)$ is a topological group. Now I think that it result is a colorally of the proposition of above, the problem is that $X=\mathbb{R}$ is Hausdorff but not compact, hence it makes me think that we can prove a similar result with local compact spaces. Any suggestion for how do I should apply the result to $\mathbb{R}$

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Juan T
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    Any group can be topologized for instance, giving it the discrete or trivial topology. But you want to do it so that the group operation, composition, is continuous. If $X$ is locally compact Hausdorff, then the composition is indeed continuous, if Homeo$(X)$ is given the compact-open topology. Munkres has a very readable proof of this fact. So the answer is yes. – Matematleta Feb 28 '21 at 03:21
  • I tried to prove that inversion was continuous, and I think that the fact that real homeomorphisms are monotone must give you an answer somehow, but I couldn't get this to work. I found this article which proves this for locally connected, locally compact Hausdorff, spaces by way of the one point compactification. By the way, I proved a general form of the continuity of composition here. – paul blart math cop Feb 28 '21 at 08:47
  • @Matematleta The issue is whether inversion is also continuous, and that's a subtle matter. It's true in the case of $\Bbb R$ though. – Henno Brandsma Feb 28 '21 at 10:28

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