Lemma

Let $c:\mathbf{I}\rightarrow X$ be a path and let W be a neighbourhood of $c$ in the space $\mathscr{C}_c(\mathbf{I}; X)$. There exists a real number $\varepsilon \in ]0,1/2]$, a neighbourhood $V_0$ of $c(0)$ and a neighbourhood $V_1$ of $c(1)$ in X having the following properties: one has $c([0, \varepsilon ])\subset V_0,c([1-\varepsilon ,1])\subset V_1$, and every path $c':\mathbf{I}\rightarrow X$ such that

$c'(t)\in V_0$ for $0\leqslant t\leqslant \varepsilon$,

(2) $c'(t) =c(t)$ for $\varepsilon \leqslant t\leqslant 1-\varepsilon$,

$c'(t)\in V_1$ for $1-\varepsilon \leqslant t\leqslant 1$,

belongs to W.

By definition of the topology of compact convergence (TG, X, p. 26, déf. 1), there exists a finite set J, a family $(U_j)_{j\in J}$ of open sets in X and a family $(K_j)_{j\in J}$ of compact subsets of $\mathbf{I}$ such that the set $W'$ of paths $c'$ satisfying $c'(K_j)\subset U_j$ for every index $j$ is a neighbourhood of $c$ contained in W. Let us then denote by $A_0$ (resp. $A_1$) the set of indices $j$ such that $0\in K_j$ (resp. $1\in K_j$); put $V_0=\bigcap_{j\in A_0}U_j$ and $V_1=\bigcap_{j\in A_1}U_j$.

Since the mapping $c$ is continuous, there exists a real number $\varepsilon \in$ $]0,1/2]$ such that $c([0, \varepsilon ])\subset V_0,c([1-\varepsilon ,1])\subset V_1,[0, \varepsilon ]\cap K_j=\emptyset$ for all $j\notin A_0$ and $[1-\varepsilon ,1]\cap K_j=\emptyset$ for all $j\notin A_1$. Let then $c'$ be a path satisfying the conditions (2). Let us prove that $c'\in W'$. Let $j\in J$ and let $t\in K_j$. If $\varepsilon \leqslant t\leqslant 1-\varepsilon ,c'(t) =c(t)$ belongs to $U_j$. If $0\leqslant t\leqslant \varepsilon ,c'(t)\in V_0$; by the choice of $\varepsilon$, we have $j\in A_0$, hence $c'(t)\in U_j$. Analogously, if $1-\varepsilon \leqslant t\leqslant 1$, we have $j\in A_1$ and $c'(t)\in V_1\subset U_j$. Thus, $c'(K_j)\subset U_j$ and $c'$ belongs to $W'$, hence to W.

Let us now prove the second assertion of Proposition 10. Suppose that the space X is locally connected by arcs. Let $c$ be a path in X and let W be a neighbourhood of $c$ in $\mathscr{C}_c(\mathbf{I}; X)$. Let $\varepsilon ,V_0$ and $V_1$ be as in the lemma. Let $T_0$ and $T_1$ be connected by arcs neighbourhoods of $c(0)$ and $c(1)$ contained in $V_0$ and $V_1$ respectively. There exists a real number $\theta$ such that $0< \theta < \varepsilon$ and such that $c([0, \theta ])\subset T_0,c([1-\theta ,1])\subset T_1$. Let $x_0\in T_0$ and $x_1\in T_1$; let $c_0$ be a path of origin $x_0$ and of term $c(\theta )$ in $T_0$ and let $c_1$ be a path of origin $x_1$ and of term $c(1-\theta )$ in $T_1$. Put

$c_0(t/\theta )$ for $0\leqslant t\leqslant \theta$,

$c'(t) =c(t)$ for $\theta \leqslant t\leqslant 1-\theta$,

$c_1((1-t)/\theta )$ for $1-\theta \leqslant t\leqslant 1$.

One thus defines a path $c'$ joining $x_0$ to $x_1$ and satisfying the conditions (2). This proves that the image of W in $X\times X$ by the mapping $(o, e)$ contains the neighbourhood $T_0\times T_1$ of $(c(0), c(1))$, whence the proposition.