Basis for Box Topology

Theorem

Let $\mathbb S = \left \langle {\left({S_i, \tau_i}\right)}\right \rangle_{i \in I}$ be a (possibly infinite) family of topological spaces where $I$ is an arbitrary index set.

Let $S$ be the cartesian product of $\mathbb S$:

$\displaystyle S := \prod_{i \in I} S_i$

Let $\mathcal B$ be the set defined as:

$\displaystyle \mathcal B = \left\{{\prod_{i \in I} U_i: U_i \in \tau_i}\right\}$

Then $\mathcal B$ is a basis for the box topology on $\mathbb S$.