Definition 2

One calls the space of loops of X, and denotes by Ω(X), the subspace of Λ(X) consisting of loops (III, p. 256, def. 1) in X.

One denotes by $\Omega_x(X)$ the set $\Lambda_{x,x}(X)$. The elements of $\Omega_x(X)$ are called loops in X at $x$ and the elements of $\varpi_{x,x}(X)$ are called classes of loops in X at $x$. The constant mapping $e_x:\mathbf{I}\rightarrow X$ with image $x$ is a loop, called the constant loop at $x$; its strict homotopy class is denoted by $\varepsilon_x$. The mapping $x\mapsto e_x$ of X into Λ(X) is continuous (III, p. 257, prop. 1).

Let X be a topological space and let $x,y,z$ be points of X. By passing to arcwise connected components, one deduces from the continuous mapping $c\mapsto \overline{c}$ of $\Lambda_{x,y}(X)$ into $\Lambda_{y,x}(X)$ (III, p. 258, corollary) a mapping of $\varpi_{x,y}(X)$ into $\varpi_{y,x}(X)$ which is denoted by $\gamma \mapsto \overline{\gamma}$. If $\gamma \in$ $\varpi_{x,y}(X),\overline{\gamma}$ is called the inverse of the path class $\gamma$.

Analogously, if one identifies the sets $\pi_0(\Lambda_{x,y}(X))\times \pi_0(\Lambda_{y,z}(X))$ and $\pi_0(\Lambda_{x,y}(X)\times \Lambda_{y,z}(X))$ (III, p. 260, Prop. 6), one deduces from the continuous mapping $(c, d)\mapsto c*d$ of $\Lambda_{x,y}(X)\times \Lambda_{y,z}(X)$ into $\Lambda_{x,z}(X)$ (III, p. 258, corollary), by passing to arcwise connected components, a mapping $C_{x,y,z}:\varpi_{x,y}(X)\times \varpi_{y,z}(X)\rightarrow \varpi_{x,z}(X)$. For $\gamma \in \varpi_{x,y}(X)$ and $\delta \in \varpi_{y,z}(X)$, the strict homotopy class $C_{x,y,z}(\gamma , \delta )$ is denoted by $\gamma \delta$. It is called the composite of the composable path classes $\gamma$ and $\delta$.

One has $\overline{\overline{\gamma}}=\gamma$ and $\overline{\gamma \delta}=\delta \gamma$.