Definition 3

Let X be a topological space. The groupoid $\varpi (X)$ is called the Poincaré groupoid, or fundamental groupoid, of the space X. Let $x$ be a point of X; the group $\pi_1(X, x)$ of classes of loops at $x$ is called the Poincaré group, or fundamental group, of the space X at the point $x$.

Let $\mathscr{U}$ be a set of subsets of X whose interiors cover X. The classes of paths in X whose image is contained in one of the subsets belonging to $\mathscr{U}$ generate the groupoid $\varpi (X)$ (lemma 4 of III, p. 272).

The orbits of the groupoid $\varpi (X)$ (II, p. 162) coincide with the arcwise connected components of the space X. In particular (loc. cit.), one has: