Content deleted Content added
Filled in 0 bare reference(s) with reFill 2 |
→Global version: fix |
||
Line 166:
{{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.)
| |||