Natural Basis of Product Topology

Theorem
Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.

Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$:
 * $\displaystyle X := \prod_{i \mathop \in I} X_i$

Then the natural basis on $X$ is the set $\BB$ of cartesian products of the form $\displaystyle \prod_{i \mathop \in I} U_i$ where:
 * for all $i \in I : U_i \in \tau_i$
 * for all but finitely many indices $i : U_i = X_i$