box topology


Let {(Xα,𝒯α)}α∈A be a family of topological spacesMathworldPlanetmath. Let Y denote the generalized Cartesian product of the sets Xα, that is

Y=∏α∈AXα.

Let ℬ denote the set of all productsPlanetmathPlanetmathPlanetmath of open sets of the corresponding spaces, that is

ℬ={∏α∈AUα|Uα∈𝒯α⁢ for all ⁢α∈A}.

Now we can construct the box product (Y,𝒮), where 𝒮, referred to as the box topology, is the topologyMathworldPlanetmath the base 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 (X0,𝒯0) and (X1,𝒯1) is (X0×X1,𝒮), where the box topology 𝒮 (which is the same as the product topology) consists of all sets of the form ⋃i∈I(Ui×Vi), where I is some index setMathworldPlanetmathPlanetmath and for each i∈I we have Ui∈𝒯0 and Vi∈𝒯1.

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