Remark 2

Let X be a topological space and let C be an arcwise connected component of X. If $x$ and $y$ are points of C, the topological space $\Lambda_{x,y}(C)$ is identified with the topological space $\Lambda_{x,y}(X)$, so that the set $\varpi_{x,y}(C)$ is identified with the set $\varpi_{x,y}(X)$. Thus, the fundamental groupoid $\varpi (C)$ is identified with the full subgroupoid of $\varpi (X)$ having C as set of points. In particular, for every point $x$ of C, the group $\pi_1(C, x)$ is identified with the group $\pi_1(X, x)$.