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!