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What is the motivation to define a manifold to be second countable? What kind of pedagogical issues does this avoid?

LASV
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    It’s not a pædagogical issue: the condition is intended to exclude such locally Euclidean spaces as the long line, which lack some otherwise highly desirable properties (like paracompactness). – Brian M. Scott Dec 01 '13 at 07:30
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    Asked here, but it only generated a few comments. –  Dec 01 '13 at 07:31
  • Ah, I see. Thank you so much @BrianM.Scott, and T.Bongers – LASV Dec 01 '13 at 07:39
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    Paritions of unity are a very useful tool to study manifolds, they allow to patch together local entities to make them global. If a manifold were non-second countable, the existence of such partitions would be not guaranteed. – Seirios Dec 01 '13 at 08:28

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