# Definition:Box Topology

Jump to navigation
Jump to search

## 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:

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

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 $\BB$.

## 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 Product Topology
- Box Topology on Finite Product Space is Product Topology

- Results about
**box topologies**can be found**here**.

### Relation between Product and Box Topology

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