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I am reading the first volume of Spivak's Differential Geometry Series, A Comprehensive Introduction to Differential Geometry and I had a question about a topic discussed on page 4.

What I am confused about is what he writes on $\sigma$ compactness. I get what the definition means, as well as his proof related to $\sigma$ compactness. My question is this: why is this relevant to the discussion of manifolds in the rest of the chapter? I think I have a rough understanding of what is going on, but I'd prefer to not make any brave assumptions.

I appreciate any help I can get on this. I'm trying to stretch myself mathematically, and I want to make sure I understand this. Thanks.

Nicholas James
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1 Answers1

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I don't have this book so I can't say for sure, so here is my guess.

Suppose you wanted to define a manifold from first principles. The obvious thing you might try is to define a manifold $M$ to be a topological space which is locally Euclidean (that is, every point has a neighborhood homeomorphic to an open subset of $\mathbb{R}^n$). Somewhat surprisingly, this condition does not imply that $M$ is Hausdorff (which implies that Hausdorffness is not a local condition!) as the line with two origins shows.

So your next attempt might be locally Euclidean + Hausdorff. Now the problem is that there are locally Euclidean Hausdorff spaces that are "too big," such as the long line, which you might regard as pathological and want to exclude from the definition of a manifold. For example, you might want every manifold to be embeddable in some $\mathbb{R}^n$ (which is true for the usual definition), and the long line is "too big" to have this property. So next you might want to impose some kind of "size condition."

The traditional size condition is to require second-countability. However, this is equivalent to requiring $\sigma$-compactness, so $\sigma$-compactness can also be used as the size condition. At this MO question you can find a discussion of other equivalent conditions, at least in the connected case.

Qiaochu Yuan
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