Proposition 8

Let $(X_i)_{i \in I}, (Y_i)_{i \in I}$ be two families of topological spaces indexed by the same set $I$. For each $i \in I$ let $f_i$ be an open mapping of $X_i$ into $Y_i$, and suppose that $f_i$ is surjective for all but a finite number of indices. Then the product mapping $f : (x_i) \to (f_i(x_i))$ of $\prod_{i \in I} X_i$ into $\prod_{i \in I} Y_i$ is open.

By virtue of Proposition 5 we need only prove that the image under $f$ of any elementary set $\prod_{i \in I} A_i$ in $\prod_{i \in I} X_i$ is open in $\prod_{i \in I} Y_i$. But this image is $\prod_{i \in I} f_i(A_i)$, and the hypotheses imply that $f_i(A_i)$ is open in $Y_i$ for each $i \in I$, and that $f_i(A_i) = Y_i$ for all but a finite number of indices; whence the result.