Let $Homeo(X)$ denote the group of homeomorphisms of a compact Hausdorff space $X$ onto itself. If $Homeo(X)$ is isomorphic to $Homeo(Y)$ (as topological groups with compact-open topology), can we conclude that $X$ is homeomorphic to $Y$ ?
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6This would imply that all so-called rigid spaces (only the identity as a homeomomorphism of itself) are homeomorphic. This is not the case... See also http://math.stackexchange.com/questions/44820/does-every-locally-compact-second-countable-space-have-a-non-trivial-automorphis – Henno Brandsma Sep 18 '13 at 14:15