Natural Basis of Product Topology/Lemma 3

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 $\family {X_i}_{i \mathop \in I}$:
 * $\displaystyle X := \prod_{i \mathop \in I} X_i$

Let $\BB$ be the set 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 $i : U_i = X_i$

Then:
 * $\displaystyle \forall B \in \BB : B = \bigcap_{j \in J} \pr_j^{-1} \sqbrk {U_j}$

where:
 * $\displaystyle B = \prod_{i \mathop \in I} U_i$
 * $J = \set{j \in I : U_i \neq X_i}$ is finite