(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (23)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

box topology

(Definition)

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

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

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

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

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

When is a 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 is some index set and for each $i\in I$ we have $U_i\in{\mathcal T}_0$ and $V_i\in{\mathcal T}_1$ .

"box topology" is owned by yark. [ full author list (2) | owner history (1) ]

(view preamble | get metadata)