TITLE: Open subsets in a manifold as submanifold of the same dimension?
QUESTION [13 upvotes]: An open set in an $n$-manifold is clearly a submanifold of the same dimension as its containing manifold (see open manifolds). 
Now, given an $n$-manifold $M$, is it true that a set, to be the underlying set of a submanifold of $M$ with dimension $n$, must be open?

REPLY [2 votes]: Let f be embedding of submanifold N into M. I only need to prove f(N) is open. Now consider Df(p) from tangent space of N at p to tangent space of M at f(p). 
Now Df(p) 1-1 (by defintion) of submanifold). But dimN=dimM. Using Rank Nullity theorem we get Df(p) 1-1 onto. Hence f is local diffeomorphism at each point. 
Hence f is diffeomorphism.(although proof of this also goes in same way as @PseudoNeo gave)
In particular f homeomorphism. Hence f(N) open.