Remark 1

Let X be a topological space and let P be a topological space reduced to a point. Identify $P\times \mathbf{I}$ with $\mathbf{I}$ by the projection pr$_2$. The set $\mathscr{C}(P\times \mathbf{I}; X)$ of homotopies between (necessarily continuous) mappings from P into X is then identified with the set $\mathscr{C}(\mathbf{I}; X)$ of paths in X. To a homotopy joining the constant mapping of image $x$ to the constant mapping of image $y$ there corresponds a path with origin $x$ and term $y$. The preceding identifications are compatible with the notions of juxtaposition and passing to the opposite (cf. III, p. 230).