nLab
Tychonoff product

Context

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic homotopy theory

Theorems

Tychonoff product

Idea

The Tychonoff product (named for Andrey Tikhonov) is simply the product in the category Top of topological spaces and continuous maps. (See also at product topological space.)

It is the usual notion of ‘product’ of topological spaces, but should be distinguished from the box product?, which is sometimes useful. (They are the same for finite products.)

Definitions

Let (X i) i(X_i)_i be a family of topological spaces. Consider the cartesian product i|X i|\prod_i {|X_i|} of the underlying sets of these

Definitions

The Tychonoff topology or product topology on i|X i|\prod_i {|X_i|} is the initial topology generated by the π i\pi_i. The Tychonoff product or topological product or simply product of the spaces X iX_i is the set i|X i|\prod_i {|X_i|} equipped with the Tychonoff product topology.

Properties

Basic

Of course, the maps π i\pi_i are continuous maps, so we have a cone in Top.

Proposition

This is in fact a product cone in TopTop.

By the general characterization of limits in Top, see here.

Proposition

The Tychonoff topology on iIX i\prod_{i \in I} X_i is equivalently the topology

  • generated from the sub-basis

    {p i 1(U i)|iI,U iτ X i} \{ p_i^{-1}(U_i)| i \in I\,, U_i \in \tau_{X_i} \}
  • generated from the basis

    { jJp j 1(U j)|JIfinitesubsetandU jτ X j}, \{ \cap_{j \in J} \p_j^{-1}(U_j) | J \subset I \; finite \; subset \; and U_j \in \tau_{X_j}\} \,,

where p i: jIX jX ip_i \colon \prod_{j \in I} X_j \to X_i are the defining projection maps.

In particular if II itself is already finite, then a basis of the product topology is given by the Cartesian products of elements of a basis in the factors.

Tychonoff’s theorem

The Tychonoff product of compact topological spaces is itself again a compact topological space. This is known as the Tychonoff theorem_, see there for more.

References

Named after A. N. Tychonoff.

Revised on April 19, 2017 14:56:13 by Urs Schreiber (92.218.150.85)