Closed set in topological product We know that $\Pi_{i=1}^{n} \overline{A_i}=\overline{A}$ where $A=\Pi_{i=1}^n A_i$ 
How we can see from this that $\Pi_{i=1}^n A_i$ i closed if and only if each $A_i$ is closed ?
thank you

If $A$ is closed, then $\overline{A}=A$, so $\prod_{i=1}^n\overline{A_i}=\prod_{i=1}^nA_i$, so $\overline{A_i}=A_i$ for each $i$ (each component is closed).
If each $A_i$ is closed, then $\overline{A}=\prod_{i=1}^n\overline{A_i}=\prod_{i=1}^nA_i=A$, so $A$ is closed, too.
