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Theorem. Every Manifold is locally compact.

This is a problem in Spivak's Differential Geometry.

However, don't know how to prove it. It gives no hints and I don't know if there is so stupidly easy way or it's really complex.

I good example is the fact that Heine Borel Theorem, I would have no clue on how to prove it if I didn't see the proof.

So can someone give me hints. I suppose if it's local, then does this imply that it's homeomorphic to some bounded subset of a Euclidean Space?

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    Hint: every point has a neighborhood homeomorphic to the open unit ball in $\mathbb R^n$.2012-01-29
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    I took the liberty of editing out your second question, which I really think you should post as a separate question since it has no direct relationship to your first question. (Also, the honest answer to "How do prove Invariance of Domain?" is "Look it up." It is certainly too hard for a nonexpert to prove as an exercise. And in fact if you google for -- invariance of domain, proof -- you will find plenty of proofs...and see that they are not so easy.)2012-01-30

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