box topology
Let
be a family of topological spaces.
Let denote the generalized Cartesian product of the sets ,
that is
Let denote the set of all products of open sets of the corresponding
spaces, that is
Now we can construct the box product , where ,
referred to as the box topology,
is the topology the base .
When 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 and
is ,
where the box topology (which is the same as the product topology)
consists of all sets of the form
,
where is some index set
and for each we have and .
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 |