Remark 2

Let X and Y be topological spaces. The canonical mapping

associates to every homotopy $\sigma : X\times \mathbf{I}\rightarrow Y$ joining two mappings $f$ and $g$ from X into Y a path with origin $f$, term $g$, in the space $\mathscr{C}_c(X; Y)$. If X is a locally compact space, this canonical mapping is bijective (TG, X, p. 28, theorem 3).

Let X be a topological space. One calls the space of paths of X, and denotes by Λ(X), the topological space $\mathscr{C}_c(\mathbf{I}; X)$ whose elements are the paths in X and whose topology is that of compact convergence (cf. TG, X, p. 27). If $x$ is a point of X, one denotes by $\Lambda_x(X)$ the subspace of Λ(X) formed by paths with origin $x$. If $y$ is a second point of X, one denotes by $\Lambda_{x,y}(X)$ the subspace of Λ(X) formed by paths with origin $x$ and term $y$.