ScalingStacks

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. ∎