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I'm not much informed about manifold but I should answer some questions about it. Based on the definition I have written an answer for the following question but I feel there is something wrong with it! Could you please help me? Q: Let M be a smooth manifold and suppose that we have an open cover for that. If S is a subset of M such that the intersection of each element of that cover and S is a submanifold, then S itself is a submanifold.

A: Fix an arbitrary point p in M. This point belongs to an element of that cover and as we know the intersection of that element with S is a submanifold, so there exists a map around p such that satisfies the condition, so we are done!

user142468
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    Note that you need to modify the hypothesis: it should say "suppose we have an open cover." Otherwise, the conclusion is clearly false. For example "a cover" could be the collection of all one-point sets in $M$, in which case the hypothesis is satisfied no matter what $S$ is. – Jack Lee Nov 21 '14 at 21:32
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    You are right. Thanks. – user142468 Nov 22 '14 at 03:55

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Yes, your proof is correct. The common way of expressing this idea is by saying "being a manifold is a local property".

Amitai Yuval
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