Corollary

For each $i \in I$ let $A_i$ be a subspace of $Y_i$. Then the topology induced on $A = \prod_{i \in I} A_i$ by the product topology on $\prod_{i \in I} Y_i$ is the product of the topologies of the subspaces $A_i$.

Let $j_i$ be the canonical injection $A_i \to Y_i$, and apply Proposition 3 to the mappings $f_i = j_i \circ \mathrm{pr}_i$ (cf. Set Theory, Chapter IV, \S\ 2, no. 4, criterion CST 14).