Definition:Box Topology

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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:

$\ds X := \prod_{i \mathop \in I} X_i$


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

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 $\BB$.

Also see

  • Results about box topologies can be found here.

Relation between Product and Box Topology