Proposition 3

Let $X$ be a set, $(Y_i)_{i \in I}$ a family of topological spaces, and for each $i \in I$ let $f_i$ be a mapping of $X$ into $Y_i$. Let $f$ be the mapping $x \to (f_i(x))$ of $X$ into $Y = \prod_{i \in I} Y_i$, and let $\mathfrak{T}$ be the coarsest topology on $X$ for which the mappings $f_i$ are continuous. Then $\mathfrak{T}$ is the inverse image under $f$ of the topology induced on $f(X)$ by the product topology on $Y$.

This is another particular case of transitivity of initial topologies (\S\ 2, no. 3, Proposition 5; cf. Set Theory, Chapter IV, \S\ 2, no. 4, criterion CST 15).