Definition 2

Let X be a topological space. One says that paths $c$ and $d$ in X are juxtaposed if one has $c(1) =d(0)$. One then calls juxtaposed path of $c$ and $d$ the path $c*d$ defined by the formula:

$c(2t)$ for $0\leqslant t\leqslant 1/2$,

$d(2t-1)$ for $1/2\leqslant t\leqslant 1$.

Its origin is that of $c$, its term that of $d$.

Let $c$ be a path in X; one calls the path opposite to $c$ and denotes by $\overline{c}$ the path defined by $\overline{c}(t) =c(1-t)$ for $t\in \mathbf{I}$.

one hasIf $cc$and$*dd=$are two juxtaposed paths in$\overline{d}*\overline{c}$. For every path $c$ in X, one hasX$,\overline{\overline{\overline{c}}d}=$and$c\overline{c.}$ are, and