0
$\begingroup$

Let $f: M \longrightarrow N$ be a smooth map between two manifolds of the same dimension, with M compact, and a regular value $y \in N$. Then the number of points in $f^{-1}(y)$ is locally constant as a function of $y$, where $y$ ranges through regular values.

I read the proof in Milnor's book "Topology from the differentiable viewpoint" but if I'm not wrong this proof concerns regular values with preimage. But I don't know how to prove it in the case where $y$ is not the preimage of some $x \in M$. In this case $f^{-1}(y)$ should be locally $0$ in a neighborhood of $y$, right?

$\endgroup$
1
$\begingroup$

Notice that the source $M$ is compact, so the image is closed. Therefore, the complement of the image in $N$ is open.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.