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  1.  
    This question -- http://mathoverflow.net/questions/72707/what-topological-manifolds-are-embeded-in-mathbbrn-closed -- 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.
  2.  

    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.

  3.  
    OK, fair enough.