Proposition 7

In a product space $\prod_{i \in I} X_i$, the closure of a product of sets $\prod_{i \in I} A_i$ is the same as the product $\prod_{i \in I} \overline{A}_i$ of their closures.

Suppose that $a = (a_i)$ lies in the closure of $\prod_{i \in I} A_i$; then for each $x \in I$, $a_x = \operatorname{pr}_x a$ is in the closure of $A_x$ because of the continuity of $\operatorname{pr}_x$ (§ 2, no. 1, Theorem 1) and therefore $a \in \prod_{i \in I} \overline{A}_i$. Conversely, let $b = (b_i) \in \prod_{i \in I} \overline{A}_i$, and let $\prod_{i \in I} V_i$ be any elementary set containing $b$; for each $i \in I$, $V_i$ contains a point $x_i \in A_i$; hence $\prod_{i \in I} V_i$ contains the point $(x_i) \in \prod_{i \in I} A_i$ and therefore $b$ lies in the closure of $\prod_{i \in I} A_i$.