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Why do we generally require that a differentiable manifold be Hausdorff and second countable? Is this universally accepted in the definition? My Professor only required the Hausdorff condition for example, but most book I have read require both. Thanks

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I don't have a good answer, but have you heard about the long line? It is the canonical example of a “manifold” which is not second countable. If you look at the wikipedia article you will see some odd behaviour when you try to put a differentiable structure on it. – Harald Hanche-Olsen Mar 25 '13 at 21:40
    
Very interesting, thank you for the response. – mike Mar 25 '13 at 21:44
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No, it’s not universal: some people study non-metrizable manifolds. This question and its answers, and the accepted answer to this question should go a long way towards answering your question. – Brian M. Scott Mar 25 '13 at 21:45
    
It is fairly common to require paracompactness instead of second-countability (it can be shown that a manifold is paracompact iff each connected component is second-countable). – Eric Wofsey Jan 14 at 7:09

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