Definition:Box Topology
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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.