Remark 1

The homomorphism $\varpi_{x,y}(f)$ is continuous if one endows the sets $\varpi_{x,y}(X)$ and $\varpi_{f(x),f(y)}(Y)$ with the quotient topology of the topology of compact convergence on $\mathscr{C}_c(\mathbf{I}; X)$ and $\mathscr{C}_c(\mathbf{I}; Y)$.

To simplify the notation, if $x,y\in X$ and $\gamma \in \varpi_{x,y}(X)$, we shall sometimes write $f_*(\gamma )$ for the element $\varpi_{x,y}(f)(\gamma )$ of $\varpi_{f(x),f(y)}(Y)$.

Let X, Y, Z be topological spaces, let $f: X\rightarrow Y,g: Y\rightarrow Z$ be continuous mappings. For every path $c$ in X, one has $(g\circ f)\circ c=$ $g\circ (f\circ c)$. It follows that one has

Let X and Y be topological spaces, let $f_0$ and $f_1$ be continuous mappings of X into Y, and let $\sigma : X\times \mathbf{I}\rightarrow Y$ be a homotopy joining $f_0$ to $f_1$. Since the mapping $(c, t)\mapsto c(t)$ of $\mathscr{C}_c(\mathbf{I}; X)\times \mathbf{I}$ into X (III, p. 257, prop. 1) is continuous, the same is true of the mapping of $\mathscr{C}_c(\mathbf{I}; X)\times \mathbf{I}\times \mathbf{I}$ into Y given by $(c, t, s)\mapsto \sigma (c(t), s)$. Consequently, the mapping $\Sigma : (c, s)\mapsto \sigma (c(\cdot ), s)$ is a continuous mapping of $\mathscr{C}_c(\mathbf{I}; X)\times \mathbf{I}$ into $\mathscr{C}_c(\mathbf{I}; Y)$ (loc. cit.). The mapping Σ is a homotopy joining the mapping $\Lambda (f_0)$ to the mapping $\Lambda (f_1)$. By restriction to the loop spaces, the mapping Σ induces a homotopy $\Omega (X)\times \mathbf{I}\rightarrow \Omega (Y)$ joining $\Omega (f_0)$ to $\Omega (f_1)$. Let $x$ be a point of X; suppose that the homotopy $\sigma$ is a homotopy pointed at $x$, and put $y=f_0(x) =f_1(x)$. The mapping Σ then induces a continuous mapping of $\Omega_x(X)\times \mathbf{I}$ into $\Omega_y(Y)$ which is a homotopy pointed at $e_x$, joining the mapping $\Omega_x(f_0)$ to the mapping $\Omega_x(f_1)$.