In the Munkres proof of countable product of second countable spaces is second countable, he stated that the set $F=\{\prod\_{i=1}^{\infty}U\_i\}$ where $U\_i=B\_i$, where $B\_i\in\mathcal{B\_i}$ for finitely many $i$'s and $\mathcal{B\_i}$ is a set of countable basis. And $U\_i = X\_i$ for the rest, where $X\_i$ is a second countable space. 

Intuitively, I understand the cardinality of $F$ should be countable. But I have trouble rigorously showing it. Any ideas?