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    This question -- -- was closed as a duplicate, but the meaning of "topological manifold" is different in the two questions, so they are not really duplicates. I propose that it be reopened. Maybe we should also ask the questioner to use a term other than "topological manifold", since that has another, more established meaning.

    The questions are the same once you know that a subspace of R^n is necessarily Hausdorff and second-countable, and the OP was informed of this in a math.SE question that he posted.

    OK, fair enough.