Why are manifolds second countable

Asked Apr 18 '17 at 18:15

Active Apr 18 '17 at 18:21

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A manifold is defined to be a second countable Hausdorff space that is locally homeomorphic to Euclidean space.

It surprises me that second countably is a requirement here, because it surprises me that a space can be locally homeomorphic to $\mathbb{E}^n$ and yet not second countable. I learned at this question that the Long Line is one such example, but I'm interested in seeing more to understand them better. What are some more examples of such spaces? What properties do they have that we might want to exclude them from the notion of "manifolds"?

edited Apr 18 '17 at 18:21

asked Apr 18 '17 at 18:15

Stella Biderman

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One other example is $\mathbb R\times X$ where $X$ is any uncountable set with the discrete topology. (I don't really know any problems that occur when this particular space is admitted though) – Milo Brandt Apr 18 '17 at 18:24
Another example: the Prüfer surface. – Hans Lundmark Apr 18 '17 at 18:46
I would say so Paracompactness holds. The implications of paracompactness are easier to see. – Apr 18 '17 at 19:15
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We want manifolds to be embeddable into $\mathbb{R}^m$ for some $m$. – Qiaochu Yuan Apr 18 '17 at 19:34

Why are manifolds second countable

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