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
Canonical name	BoxTopology
Date of creation	2013-03-22 12:46:55
Last modified on	2013-03-22 12:46:55
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	9
Author	yark (2760)
Entry type	Definition
Classification	msc 54A99
Synonym	box product topology
Related topic	ProductTopology
Defines	box product