Lemma 7.1.16 (0P8P) · ScalingStacks

0P8P

Lemma 7.1.16. Let $E$ be a finite subset of $X$ and $\gamma$ a path in $X$ . Let $B$ be the set of connected components $I$ of $[0,1]\setminus\gamma^{-1}(E)$ such that $\gamma_{|\bar{I}}$ is not nullhomotopic. Then $B$ is finite and there are paths $\gamma^{\prime}$ and $\gamma^{\prime\prime}$ homotopic to $\gamma$ such that

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$\gamma$ and $\gamma^{\prime}$ coincide on $\bigcup_{I\in B}\bar{I}$ and $\gamma^{\prime}([0,1]\setminus\bigcup_{I\in B}\bar{I})\subset E$
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$\gamma^{\prime\prime-1}(E)$ is finite.

0P8Q

Proof. Let ${\mathcal{U}}$ be an open covering of $X$ by connected and simply connected subsets, each of which contain at most one element of $E$ . By Lebesgue’s number Lemma, there are only finitely many $I\in\pi_{0}([0,1]\setminus\gamma^{-1}(E))$ such that $\bar{I}$ is not contained in an element of $\gamma^{-1}({\mathcal{U}})$ . So, $B$ is finite.

We can write $\gamma$ as a finite composition of its restrictions to $\bar{I}$ for $I\in B$ interlaced with finitely many paths that satisfy the assumptions of Lemma 7.1.15. Thanks to that lemma, we obtain a path $\gamma^{\prime}$ satisfying the requirements of the lemma. By shrinking the intervals on which $\gamma^{\prime}$ is constant to points, we obtain a path $\gamma^{\prime\prime}$ as desired. ∎