# Definition:Box Topology

## Definition

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

Let $S$ be the cartesian product of $\left \langle {S_i} \right \rangle_{i \mathop \in I}$, that is:

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

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 $S$, as shown on Basis for Box Topology.

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

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

## Also see

• Results about box topologies can be found here.