Remark 1

Let X be a topological space. Let $n$ be an integer $\geqslant 1$ and let $(c_1, . . . , c_n)$ be a sequence of paths in X such that $c_i$ and $c_{i+1}$ are juxtaposable for $1\leqslant i\leqslant n-1$ (such a sequence is called a sequence of juxtaposable paths). Let us denote by $c$ the path

and by $c'$ the path defined by $c'(t) =c_i(nt-i+ 1)$ for $1\leqslant i\leqslant n$ and $t\in [\frac{i-1}{n},\frac{i}{n}]$. The paths $c$ and $c'$ have the same image and are strictly homotopic: one is the composite of the other with a homeomorphism of $\mathbf{I}$ leaving 0 and 1 fixed (cf. III, p. 289, example). We shall sometimes denote by $c_1*c_2* \cdots *c_n$ the path $c'$.

There exists a unique oriented graph $\varpi (X)$ whose set of vertices is X and whose set of arrows joining a point $x$ to a point $y$ is $\varpi_{x,y}(X)$, and in which the mappings $C_{x,y,z}$ define a law of composition. By proposition $1,\varpi (X)$ is a groupoid (II, p. 162, def. 4). For every $x\in X$, the identity element at the vertex $x$ of this groupoid is the class of the constant loop with image $x$. The inverse of an arrow $\gamma$ is the arrow $\overline{\gamma}$, which we shall also denote by $\gamma^{-1}$. In particular, the law of composition $C_{x,x,x}$ endows, for every $x\in X$, the set $\varpi_{x,x}(X)$ with a group structure; this group is denoted by $\pi_1(X, x)$.