Remark 3.5.7.9. In the situation of Proposition 3.5.7.7, it is not necessary to verify the vanishing of the group $\pi _{m}(X,x)$ for every choice of vertex $x \in X$; it is enough to check this at one point from each connected component of $X$ (see Example 3.2.2.18). In particular, if $X$ is connected, then it is enough to check that this condition holds for any choice of vertex $x \in X$.
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