Proposition 10

Let X be a topological space. The mapping $(o, e)$ of $\Lambda (X) =\mathscr{C}_c(\mathbf{I}; X)$ into $X\times X$, which associates with a path $c$ in X the pair $(c(0), c(1))$, is continuous. If the space X is locally arcwise connected, this mapping is open.

We already know that the origin mappings $o$ and term $e$ of Λ(X) into X are continuous (III, p. 257), whence the first assertion.