Abstract Nonsense

Local Homeo(Diffeo)morphisms to Global Homeo(Diffeo)morphisms

Point of Post: In this post we discuss an important consequence of the inverse function theorem which relates local diffeomorphisms to global diffeomorphisms.

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Local to Global Diffeomorphism

Let $U\subseteq\mathbb{R}^n$ be open and $f:U\to\mathbb{R}^m$. We call $f$ a local diffeomorphism if for every $x\in U$ there exists a neighborhood $V$ of $x$ such that $f(V)$ is open in $\mathbb{R}^m$ and the map $f_{\mid V}:V\to f(V)$ is a diffeomorphism. We define local homeomorphisms similarly. We note the obvious theorem
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Theorem: Let $U\subseteq\mathbb{R}^n$ be open. Then, $f:U\to\mathbb{R}^m$ is a homeo(diffeo)morphism if and only if it’s a bijective local homeo(diffeo)morphism.

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Which is true by definition since differentiability is a local property. The important fact we’d like to note is the following:

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Theorem: Let $f:U\to\mathbb{R}^m$ be a local homeomorphism, then $f$ is an open map.

Proof: Let $V\subseteq U$ be open. We know that for each $x\in V$ there is neighborhood $O_x\subseteq U$ for which $f(O_x)$ is open in $\mathbb{R}^m$ and $f_{\mid O_x}:O_x\to f(O_x)$ is a homeomorphism. Note then that

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$\displaystyle f(V)=f\left(\bigcup_{x\in V}\left(O_x\cap V\right)\right)=\bigcup_{x\in V}f(O_x\cap V)$

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But, since $O_x\cap V$ is open in $O_x$ and $f$ is a homeomorphism we have that $f(O_x\cap V)$ is open in $O_x$, but since $O_x$ is open in $\mathbb{R}^m$ this implies that $f(O_x\cap V)$ is also open in $\mathbb{R}^m$ and so $f(V)$ is the union of open sets in $\mathbb{R}^m$ and so trivially open. $\blacksquare$

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So, what does this have to do with the inverse function theorem? Well, the following:

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Theorem: Let $f:U\to\mathbb{R}^n$, with $U\subseteq\mathbb{R}^n$ open, be such that the total derivative $D_f(x_0)$ is invertible for each $x_0\in U$. Then, $f$ is a local diffeomorphism and so, in particular, open.

Proof: This is an immediate consequence of the inverse function theorem which says that since $D_f(x_0)$ is invertible $x_0$ has a neighborhood for which the restriction of $f$ to this neighborhood is a diffeomorphism. $\blacksquare$

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References:

1.  Spivak, Michael. Calculus on Manifolds; a Modern Approach to Classical Theorems of Advanced Calculus. New York: W.A. Benjamin, 1965. Print.

2. Apostol, Tom M. Mathematical Analysis. Reading, MA: Addison-Wesley Pub., 1974. Print.