Product Space Basis Induced from Factor Space Bases

Theorem
Let $\family{\struct{S_\alpha, \tau_\alpha}}_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$.

Let $\BB_\alpha$ be a basis for the topology $\tau_\alpha$ for each $\alpha \in I$.

Let $\displaystyle \struct{S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct{S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$.