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Is every locally compact, Hausdorff, locally path-connected topological group $G$ locally Euclidean? (That would imply of course also being a Lie group). Is it true when countable basis is assumed? I wasn't able to find a discussion of this question in the literature on topological groups and the Hilbert 5th problem.

EDIT: As YCor rightly pointed out, let's assume that that $G$ is of finite topological dimension. The general theme of my question is to identify a minimal set of conditions on topological group making it into a Lie group.

Is every locally compact, Hausdorff, locally path-connected topological group $G$ locally Euclidean? (That would imply of course also being a Lie group.) Is it true when countable basis is assumed? I wasn't able to find a discussion of this question in the literature on topological groups and the Hilbert 5th problem.

EDIT: As YCor rightly pointed out, let's assume that $G$ is of finite topological dimension. The general theme of my question is to identify a minimal set of conditions on a topological group making it into a Lie group.

Is every locally compact, Hausdorff, locally path-connected topological group locally Euclidean? (That would imply of course also being a Lie group). Is it true when countable basis assumption is added? I wasn't able to find a discussion of this question in the literature on topological groups and the Hilbert 5th problem.

Is every locally compact, Hausdorff, locally path-connected topological group $G$ locally Euclidean? (That would imply of course also being a Lie group). Is it true when countable basis is assumed? I wasn't able to find a discussion of this question in the literature on topological groups and the Hilbert 5th problem.

EDIT: As YCor rightly pointed out, let's assume that that $G$ is of finite topological dimension. The general theme of my question is to identify a minimal set of conditions on topological group making it into a Lie group.