# box topology

Let $\{(X_{\alpha},{\mathcal{T}}_{\alpha})\}_{\alpha\in A}$ be a family of topological spaces. Let $Y$ denote the generalized Cartesian product of the sets $X_{\alpha}$, that is

 $Y=\prod_{\alpha\in A}X_{\alpha}.$

Let ${\mathcal{B}}$ denote the set of all products of open sets of the corresponding spaces, that is

 ${\mathcal{B}}=\left\{\prod_{\alpha\in A}U_{\alpha}\,\Biggm{|}\,U_{\alpha}\in{% \mathcal{T}}_{\alpha}\text{ for all }\alpha\in A\right\}.$

Now we can construct the box product $(Y,{\mathcal{S}})$, where ${\mathcal{S}}$, referred to as the box topology, is the topology the base ${\mathcal{B}}$.

When $A$ is a finite (http://planetmath.org/Finite) set, the box topology coincides with the product topology.

## Example

As an example, the box product of two topological spaces $(X_{0},{\mathcal{T}}_{0})$ and $(X_{1},{\mathcal{T}}_{1})$ is $(X_{0}\times X_{1},{\mathcal{S}})$, where the box topology ${\mathcal{S}}$ (which is the same as the product topology) consists of all sets of the form $\bigcup_{i\in I}(U_{i}\times V_{i})$, where $I$ is some index set and for each $i\in I$ we have $U_{i}\in{\mathcal{T}}_{0}$ and $V_{i}\in{\mathcal{T}}_{1}$.

Title box topology BoxTopology 2013-03-22 12:46:55 2013-03-22 12:46:55 yark (2760) yark (2760) 9 yark (2760) Definition msc 54A99 box product topology ProductTopology box product