Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Could you please explain the following statement:

Let $X,Y$ be manifolds, and assume that $\bar{X}$ is compact, and let $\phi:\bar{X} \rightarrow Y$. Suppose $y_0 \in Y-\phi(\partial X)$ is a regular value of $\phi$. Then from the inverse function theorem and compactness, $\phi^{-1}(y_0)$ is finite.

Why is it finite? I don't get it from the theorem, which talks about local diffeomorphisms?

share|cite|improve this question
up vote 3 down vote accepted

Assuming $\dim(X)=\dim(Y)$, around any point $x\in\phi^{-1}(y_0)$ there is an open set $U_x\subset X$ such that $\phi_{|U_x}:U_x\to\phi(U_x)$ is a diffeomorphism (that's what the inverse function theorem gives you). The point is that $\phi_{|U_x}$ is injective, so the open set $U_x$ doesn't contain another $x'\in\phi^{-1}(y_0)$, hence $\phi^{-1}(y_0)$ is discrete. Since it's also closed, and $\overline X$ is compact, it must be finite.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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