## Inverse product mapping

In my book, it states that:

$\prod _{ \alpha \in A } E_ \alpha = \bigcap _{ \alpha \in A} \prod _{ \alpha \in A }^{-1} E_ \alpha$, but did not explain the reason.

I'm really lost on this matter, I know that $\prod ^{-1} _{ \alpha \in A} E = E \times \prod _ { \beta \neq \alpha } X_ \beta$, for $E \subset X$, X a metric space, but how does this statement works?

Thank you!