The problem is indeed an exercise in spivak's 'a comprehensive introduction to differential geometry' vol 1, chap 2, exercise 24.
I can do 24(a) and a half part of (b), but got stuck on the other half....
I first gave the result of 24(a), you can just use it:
If M is a manifold, a set in M can be made into a k-dimensional submanifold of M if and only if around each point in there is a coordinate system on M such that .
Now here is the second part of the question which I got stuck:
If the subset can be made into a closed submanifold, then the coordinate systems mentioned above exist around every point of M (not just ).
My idea is that suppose M is connected, then let , where U is the coordinates mentioned above, then W is an also a submanifold in M containing , W is open obviously, I want to show W is also closed so that W=M, then we are done. But I got stuck here, maybe my idea was wrong.....
Please note that in Spivak's book, a closed submanifold in M is defined as:
(a) an immersed submanifold whose inclusion map is also an embedding, and:
(b) the subset is a closed subset of M.
Namely, a submanifold is just an embedding submanifold, and a closed submanifold is an embedding submanifold with itself is a closed set in the whole space.
Thank you in advance! Hope some guys who are familar with Spivak's book can help me~