Definition:Box Topology

Definition
Let $\left \langle {\left({X_i, \mathcal T_i}\right)} \right \rangle_{i \in I}$ be an |$I$-indexed family of topological spaces.

Let $X$ be the cartesian product of $\left \langle {X_i} \right \rangle_{i \in I}$, i.e.:
 * $\displaystyle X := \prod_{i \mathop \in I} X_i$

Define:
 * $\displaystyle \mathcal B := \left\{{\prod_{i \mathop \in I} U_i: \forall i \in I: U_i \in \mathcal T_i}\right\}$

Then $\mathcal B$ is a synthetic basis on $X$, as shown on Basis for Box Topology.

The box topology on $X$ is defined as the topology $\mathcal T$ generated by the synthetic basis $\mathcal B$.

The topological space $\left({X, \mathcal T}\right)$ is called the box product of $\left \langle {\left({X_i, \mathcal T_i}\right)} \right \rangle_{i \in I}$.

Also see

 * Basis for Box Topology
 * Tychonoff Topology