Remark 3

Let X be a topological space and let $x,y,z$ be points of X. The mapping $(c, d)\mapsto c*d$ of $\Lambda_{x,y}(X)\times \Lambda_{y,z}(X)$ into $\Lambda_{x,z}(X)$ is continuous (III, p. 258, corollary of Prop. 2). It should be noted that the composition mapping $\varpi_{x,y}(X)\times \varpi_{y,z}(X)\rightarrow \varpi_{x,z}(X)$ deduced from it is not necessarily continuous when the sets $\varpi_{x,y}(X),\varpi_{y,z}(X)$ and $\varpi_{x,z}(X)$ are endowed with the quotient topologies (cf. TG, I, p. 35 and III, p. 259). However, for every $\gamma_0\in \varpi_{x,y}(X)$ and every $\delta_0\in \varpi_{y,z}(X)$, the partial mappings $\gamma \mapsto \gamma \delta_0$ of $\varpi_{x,y}(X)$ into $\varpi_{x,z}(X)$ and $\delta \mapsto \gamma_0\delta$ of $\varpi_{y,z}(X)$ into $\varpi_{x,z}(X)$ are homeomorphisms. In fact, let $c_0\in \Lambda_{x,y}(X)$ be a path of class $\gamma_0$. The mapping $d\mapsto c_0*d$ is a continuous mapping of $\Lambda_{y,z}(X)$ into $\Lambda_{x,z}(X)$. The mapping $\delta \mapsto \gamma_0\delta$ is deduced from it by passing to quotients, and is therefore continuous. The same holds for the mapping $\delta '\mapsto \gamma_0^{-1}\delta '$ of $\varpi_{x,z}(X)$ into $\varpi_{y,z}(X)$. Since these two mappings are inverse to one another, they are homeomorphisms. One reasons analogously for the mapping $\gamma \mapsto \gamma \delta_0$. See also IV, p. 374.