nLab
box topology

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

Given a set {X i} iI\{X_i\}_{i \in I} of topological spaces, then the box topology on the Cartesian product iIX i\underset{i \in I}{\prod} X_i of the underlying sets of these spaces is the topology which is generated from the topological base whose elements are the Cartesian products

iIU iiIX i \underset{i \in I}{\prod} U_i \;\subset\; \underset{i\in I}{\prod} X_i

of open subsets U iX iU_i \subset X_i of each of the factor spaces.

Properties

Relation to Tychonoff topology

If the index set II is a finite set, then this box topology coincides with the Tychonoff topology on the Cartesian product. For general II however the Tychnoff topology has as base open subsets only those products iIU i\underset{i \in I}{\prod} U_i for which all but a finite number of factors are in fact the corresponding total space X iX_i.

Hence for non-finite index set II the box topology is strictly finer that the Tychonoff topology. Beware that it is the Tychonoff toopology which yields the actual Cartesian product in the category Top of topological spaces. Accordingly, for non-finite II the box topology fails to possess the properties that one expects from a categorical product.

For example, there may be a family of continuous functions, f i:XX if_i: X \to X_i, for which the associated map by components, XiIX iX \to \underset{i\in I}{\prod} X_i, is not continuous for the box topology.

References

Last revised on May 4, 2017 at 12:31:54. See the history of this page for a list of all contributions to it.