Proposition 1

Let X and Z be topological spaces. For a mapping $\varphi$ of Z into the space of paths $\mathscr{C}_c(\mathbf{I}; X)$ to be continuous, it is necessary and sufficient that the mapping $(z, t)\mapsto \varphi (z)(t)$ be a continuous mapping of $Z\times \mathbf{I}$ into X. In particular, the mapping $(c, t)\mapsto c(t)$ of $\mathscr{C}_c(\mathbf{I}; X)\times \mathbf{I}$ into X is continuous.

The topological space $\mathbf{I}$ being locally compact, the proposition follows from TG, X, p. 28, th. 3.

We denote $o$ and call origin mapping the mapping $c\mapsto c(0)$ of Λ(X) into X; we denote $e$ and call term mapping the mapping $c\mapsto c(1)$ of Λ(X) into X. The mappings $o$ and $e$ are continuous (TG, X, p. 27, remark 1). The pairs of juxtaposed paths in X are the elements of the fibered product of the X-spaces $(\Lambda (X), e)$ and $(\Lambda (X), o)$.