Proposition 4

Let $X_1, X_2$ be two topological spaces; then for each $a_1 \in X_1$, the mapping $x_2 \to (a_1, x_2)$ is a homeomorphism of $X_2$ onto the subspace $\{a_1\} \times X_2$ of $X_1 \times X_2$.

This is a particular case of Corollary 1 of Proposition 1 of no. 1 applied to the constant function $x_2 \to a_1$.

The mapping $x_2 \to (a_1, x_2)$ is a continuous section (\S\ 3, no. 5) with respect to the equivalence relation $\mathrm{pr}_2 z = \mathrm{pr}_2 z'$ in $X_1 \times X_2$; the quotient space $X_1 \times X_2$ by this equivalence relation is therefore homeomorphic to $X_2$.