I have a exercise, but the exercise isn't written very well. Do you understand, what the statement should be?:
"M is a smooth submanifold of N iff every point has an open nbh , s.t. is a smooth submanifold of U".
It doesn't make sense to me, because p is a point in M and U_p a subset in N. And what about the set U?
Our definition of a submanifold: M is a submanifold of N if there exist a smooth embedding and immersion f:M-> N
The last part of the statement should be "is a smooth submanifold of N", not "U".
Ok. Thank you. But what about the set ? M isn't a subset of N.
Does he identify the set M with f(M), i.e. ?
It would seem to me that the author is making the identification you suggest (or the author simply made a typo and meant to write f(M) ).
In my notes i have read, that:
Let f:M->N is an injective immersion
Then f is a topological embedding <=> f is a local topological embedding, i.e. for each there is a nbh. s.t. is hom÷omorph to f(U_p).
Therefore i want to proof this statement. The direction => is easy. How can i show the other direction:
we know f:M->f(M) is bijective. Why it is continuous?