Does there exist a Hausdorff topological group which is contractible and of finite covering dimension but which is not homeomorphic to $\mathbb{R}^n$ for some $n$?
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If a topological group is contractible, then it is locally contractible (using the group operation produce a contraction which does not move the unit of the group). By a classical result of [A. Gleason, R. Palais, On a class of transformation groups, Amer. J. Math. 79 (1957), 631–648], a locally pathfinite finitedimensional topological group is a Lie group and being contractible, is homeomorphic to an Euclidean space. 

