If $\{X\_j:j\in J\}$ is a family of topological spaces and $A\_j\subseteq X\_j$, is it true that $\displaystyle\overline{\Pi\_{j\in J}A\_j}=\displaystyle{\Pi\_{j\in J}\overline{A\_j}}$? Is there an easy way to prove this?

Of course, we are considering in $\displaystyle{\Pi\_{j\in J}X\_j}$ the product topology.

Thanks.