Lemma 7.1.4 (0P88) · ScalingStacks

Lemma 7.1.4. The following conditions are equivalent:

(1)

there is a finite subset $E_{1}$ of $X$ such that $f(X-E_{1})$ is open in $X^{\prime}$ and $f_{|X-E_{1}}:X-E_{1}\to f(X-E_{1})$ is a homeomorphism
(2)

$X_{f}$ is finite
(3)

there is a finite subset $E_{2}$ of $X$ such that $f_{|X-E_{2}}:X-E_{2}\to f(X-E_{2})$ is a homeomorphism
(4)

given $x\in X$ , there is a finite subset $E_{x}$ of $X-\{x\}$ such that $f_{|X-E_{x}}$ is injective
(5)

there is a finite subset $E_{3}$ of $X$ such that $f_{|X-E_{3}}$ is injective.

Proof. The implication $(1)\Rightarrow(2)$ follows from the fact that $X_{f}\subset f^{-1}(f(E_{1}))$ . For the implication $(2)\Rightarrow(3)$ , take $E_{2}=X_{f}$ . For $(3)\Rightarrow(4)$ , take $E_{x}=(X-\{x\})\cap(f^{-1}(f(x))\cup E_{2})$ . The implication $(4)\Rightarrow(5)$ is immediate.

Let us show that $(5)\Rightarrow(1)$ . Note first that an injective continuous map ${\mathbf{R}}\to{\mathbf{R}}$ is open and a homeomorphism onto its image. It follows that the implication holds when $X$ and $X^{\prime}$ are homeomorphic to ${\mathbf{R}}$ and $E_{3}=\emptyset$ .

Consider now the general case. There is a finite subset $E_{1}$ of $X$ containing $E_{3}$ such that $X-E_{1}$ and $X^{\prime}-f(E_{1})$ are homeomorphic to a finite disjoint union of copies of ${\mathbf{R}}$ . By the discussion above, the restriction of $f$ to a connected component of $X-E_{1}$ is open and a homeomorphism onto its image, so the same holds for $f_{|X-E_{1}}$ .

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