Definition of topological manifold

Question

This might be a stupid question, but I was wondering why we define the topological manifold to be Hausdorff and Second countable?

Thanks :)

See this question and its answers for Hausdorffness. – Brian M. Scott Oct 15 '14 at 01:57 — Brian M. Scott, Oct 15 '14 at 01:57
... and 2nd countability. – Moishe Kohan Oct 15 '14 at 15:48 — Moishe Kohan, Oct 15 '14 at 15:48

Seth · Accepted Answer · 2014-10-15T03:49:25.790

The key property of a manifold is that it is locally euclidean. However, another desired property is that it is embeddable in euclidean space. Since Hausdorff and second countable are inherited by subspaces, for a manifold to be embeddable in euclidean space it must satisfy both conditions.

Conversely, it has been proven that any locally euclidean, Hausdorff, second countable space can be embedded in euclidean space.

score 1 · Answer 2 · answered Oct 15 '14 at 02:00

1

Otherwise you get some strange stuff like the "long line."

answered Oct 15 '14 at 02:00

ncmathsadist

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Definition of topological manifold

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