# Application of inverse function theorem geometry

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?

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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.