\begin{definition}[Definition:Box Topology]
Let $\family {\struct {X_i, \tau_i}}_{i \mathop \in I}$ be an $I$-indexed family of topological spaces.
Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$, that is:
:$\ds X := \prod_{i \mathop \in I} X_i$
Define:
:$\ds \BB := \set {\prod_{i \mathop \in I} U_i: \forall i \in I: U_i \in \tau_i}$
Then $\BB$ is a synthetic basis on $X$, as shown on Basis for Box Topology.
The '''box topology''' on $X$ is defined as the topology $\tau$ generated by the synthetic basis $\BB$.
\end{definition}