Example

Let $c$ be a path in X and let $\varphi :\mathbf{I}\rightarrow \mathbf{I}$ be a continuous mapping such that $\varphi (0) = 0$ and $\varphi (1) = 1$. The paths $c$ and $c\circ \varphi$ are strictly homotopic. In fact, the mapping $\sigma :\mathbf{I}\times \mathbf{I}\rightarrow X$ defined by $\sigma (t, s) =c((1-s)t+s\varphi (t))$ is a strict homotopy joining $c$ to $c\circ \varphi$.

Let X be a topological space. Recall (cf. III, p. 257) that Λ(X) denotes the topological space $\mathscr{C}_c(\mathbf{I}; X)$ of paths in X and that for $x$, $y\in X$, $\Lambda_{x,y}(X)$ is the subspace of Λ(X) formed by the paths with origin $x$ and end $y$. The family of sets $\Lambda_{x,y}(X)$, for $x,y\in X$, is a partition of the space of paths in X. By the canonical bijection (III, p. 257, remark 2) of $\mathscr{C}(\mathbf{I}\times \mathbf{I}; X)$ onto $\mathscr{C}(\mathbf{I}; \Lambda (X))$, strict homotopies correspond to paths $c:\mathbf{I}\rightarrow \Lambda (X)$ whose image is contained in a subspace of the form $\Lambda_{x,y}(X)$. The relation “the paths $c_0$ and $c_1$ are strictly homotopic” is therefore an equivalence relation in Λ(X) (III, p. 259, prop. 5) and the equivalence classes for this relation are the arcwise connected components of the subspaces $\Lambda_{x,y}(X)$ of the space of paths in X. One denotes by $\varpi_{x,y}(X)$ the set $\pi_0(\Lambda_{x,y}(X))$ and calls every element of $\varpi_{x,y}(X)$ a class of paths joining $x$ to $y$.