Proposition 2

Let X be a topological space. The map which associates with a path $c$ the opposite path $\overline{c}$ is a homeomorphism of Λ(X) onto itself. The juxtaposition of paths $(c, d)\mapsto c*d$ is a homeomorphism of the space $(\Lambda (X), e)\times_X(\Lambda (X), o)$ onto Λ(X).

The mapping $t\mapsto 1-t$ is a homeomorphism of the space $\mathbf{I}$ onto itself; the first assertion follows.

Let C denote the fibered product $(\Lambda (X), e)\times_X(\Lambda (X), o)$. Let $\gamma : C\rightarrow$ Λ(X) be the mapping $(c, d)\mapsto c*d$ and $\delta : C\times \mathbf{I}\rightarrow$ X the mapping $((c, d), t)\mapsto (c*d)(t)$. The restrictions of the mapping $\delta$ to $C\times [0,\frac{1}{2}]$ and $C\times [\frac{1}{2},1]$ are continuous by virtue of formula (1) and of prop. 1. The mapping $\delta$ is therefore continuous (TG, I, p. 19, prop. 4), as is the mapping $\gamma$ (prop. 1). We have $c(t) = (c*d)(\frac{t}{2})$ and $d(t) = (c*d)(\frac{1+t}{2})$, hence $\gamma$ is injective.

Let $g$ be a path in X; for $t\in \mathbf{I}$, put

$c_g(t) =g(\frac{t}{2})$ and $d_g(t) =g(\frac{1 + t}{2})$.

The mappings $c_g$ and $d_g$ thus defined are juxtaposed paths in X and we have $c_g*d_g=g$. Moreover, the mappings $g\mapsto c_g$ and $g\mapsto d_g$ are continuous mappings of the space Λ(X) into itself (prop. 1). It follows that the mapping $\gamma$ is a homeomorphism.