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 $\displaystyle C^\infty$ manifold, a set $\displaystyle M_1$ in M can be made into a k-dimensional submanifold of M if and only if around each point in $\displaystyle M_1$ there is a coordinate system $\displaystyle (x,U)$ on M such that $\displaystyle M_1 \cap U=\{p:\quad x^{k+1}(p)=\cdots=x^n(p)=0\}$.

Now here is the second part of the question which I got stuck:

If the subset $\displaystyle M_1$ can be made into a closed submanifold, then the coordinate systems mentioned above exist around every point of M (not just $\displaystyle M_1$).

My idea is that suppose M is connected, then let $\displaystyle W= \bigcup U$, where U is the coordinates mentioned above, then W is an also a submanifold in M containing $\displaystyle M_1$, 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 $\displaystyle M_1$ in M is defined as:

(a) an immersed submanifold whose inclusion map is also an embedding, and:

(b) the subset $\displaystyle M_1$ 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~