Definition:Box Topology

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

Define:
 * $\displaystyle \BB := \set {\prod_{i \mathop \in I} U_i: \forall i \in I: U_i \in \tau_i}$

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 $\BB$.

Also see

 * Basis for Box Topology
 * Box Topology may not form Categorical Product in the Category of Topological Spaces
 * Box Topology may not be Coarsest Topology such that Projections are Continuous
 * Box Topology contains Tychonoff Topology
 * Box Topology on Finite Product Space is Tychonoff Topology

Relation between Box and Tychonoff Topology

 * Results about the relation between the box topology and the Tychonoff topology can be found here.