Corollary

A product $\prod_{i \in I} A_i$ of non-empty sets is closed in the product space $\prod_{i \in I} X_i$ if and only if $A_i$ is closed in $X_i$ for each $i \in I$.

If $I$ is finite, a product $\prod_{i \in I} A_i$ is open provided that $A_i$ is open in $X_i$ for each $i \in I$; but this is no longer so if $I$ is infinite.