why f(M) is sub manifold

Question

Let map f of M into N be an injective immersion. show taht if M is compact then f(M) is submanifold of N.

What have you tried? Where do you know/suppose the compactness assumption will be important? — zibadawa timmy, Nov 12 '13 at 06:37

score 0 · Answer 1 · answered Apr 22 '21 at 06:46

Provided f is an injective continuous mapping, we can know that f is a topological embedding from the fact that M is compact and N is a Hausdorff space. Then, f is a constant-rank embedding, which shows that f(M) is exactly a submanifold of N by Constant Rank Theorem.

why f(M) is sub manifold

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