Editing Inverse function theorem (section)

== Global version ==
The inverse function theorem is a local result; it applies to each point. ''A priori'', the theorem thus only shows the function <math>f</math> is locally bijective (or locally diffeomorphic of some class). The next topological lemma can be used to upgrade local injectivity to injectivity that is global to some extent.

{{math_theorem|name=Lemma|math_statement=<ref>One of Spivak's books (Editorial note: give the exact location).</ref>{{full citation needed|date=August 2023}}<ref>{{harvnb|Hirsch|1976|loc=Ch. 2, § 1., Exercise 7.}} NB: This one is for a <math>C^1</math>-immersion.</ref> If <math>A</math> is a closed subset of a (second-countable) topological manifold <math>X</math> (or, more generally, a topological space admitting an [[exhaustion by compact subsets]]) and <math>f : X \to Z</math>, <math>Z</math> some topological space, is a local homeomorphism that is injective on <math>A</math>, then <math>f</math> is injective on some neighborhood of <math>A</math>.}}

Proof:<ref>Lemma 13.3.3. of [https://www.utsc.utoronto.ca/people/kupers/wp-content/uploads/sites/50/2020/12/difffop-2020.pdf Lectures on differential topology] utoronto.ca</ref> First assume <math>X</math> is [[compact space|compact]]. If the conclusion of the theorem is false, we can find two sequences <math>x_i \ne y_i</math> such that <math>f(x_i) = f(y_i)</math> and <math>x_i, y_i</math> each converge to some points <math>x, y</math> in <math>A</math>. Since <math>f</math> is injective on <math>A</math>, <math>x = y</math>. Now, if <math>i</math> is large enough, <math>x_i, y_i</math> are in a neighborhood of <math>x = y</math> where <math>f</math> is injective; thus, <math>x_i = y_i</math>, a contradiction.

In general, consider the set <math>E = \{ (x, y) \in X^2 \mid x \ne y, f(x) = f(y) \}</math>. It is disjoint from <math>S \times S</math> for any subset <math>S \subset X</math> where <math>f</math> is injective. Let <math>X_1 \subset X_2 \subset \cdots </math> be an increasing sequence of compact subsets with union <math>X</math> and with <math>X_i</math> contained in the interior of <math>X_{i+1}</math>. Then, by the first part of the proof, for each <math>i</math>, we can find a neighborhood <math>U_i</math> of <math>A \cap X_i</math> such that <math>U_i^2 \subset X^2 - E</math>. Then <math>U = \bigcup_i U_i</math> has the required property. <math>\square</math> (See also <ref>Dan Ramras (https://mathoverflow.net/users/4042/dan-ramras), On a proof of the existence of tubular neighborhoods., URL (version: 2017-04-13): https://mathoverflow.net/q/58124</ref> for an alternative approach.)

The lemma implies the following (a sort of) global version of the inverse function theorem:

{{math_theorem|name=Inverse function theorem|math_statement=<ref>Ch. I., § 3, Exercise 10. and § 8, Exercise 14. in V. Guillemin, A. Pollack. "Differential Topology". Prentice-Hall Inc., 1974. ISBN 0-13-212605-2.</ref> Let <math>f : U \to V</math> be a map between open subsets of <math>\mathbb{R}^n</math> or more generally of manifolds. Assume <math>f</math> is continuously differentiable (or is <math>C^k</math>). If <math>f</math> is injective on a closed subset <math>A \subset U</math> and if the Jacobian matrix of <math>f</math> is invertible at each point of <math>A</math>, then <math>f</math> is injective on a neighborhood <math>A'</math> of <math>A</math> and <math>f^{-1} : f(A') \to A'</math> is continuously differentiable (or is <math>C^k</math>).}}

Note that if <math>A</math> is a point, then the above is the usual inverse function theorem.