
Submanifold
Hello,
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 $\displaystyle p \in M$ has an open nbh $\displaystyle U_p \subset N$ , s.t. $\displaystyle U_p \cap M$ 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
thank you

The last part of the statement should be "is a smooth submanifold of N", not "U".

Ok. Thank you. But what about the set $\displaystyle U_p$? M isn't a subset of N.
Does he identify the set M with f(M), i.e. $\displaystyle U_p \cap M= U_p \cap f(M)$?
Regards

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) ).

Hello,
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 $\displaystyle p \in M$ there is a nbh. $\displaystyle U_p$ s.t. $\displaystyle U_p$ is homöomorph 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?
Regards