nLab
Stone-Cech compactification

Contents

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

The forgetful functor/full and faithful subcategory embedding from compact Hausdorff topological spaces into all topological spaces

U:Top CHausTop U \;\colon\; Top_{CHaus} \hookrightarrow Top

has a left adjoint

β:TopTop CHaus \beta \;\colon\; Top \longrightarrow Top_{CHaus}

which sends a general topological space to a compact Hausdorff topological space, called its Stone-Čech compactification. This hence exhibits Top CHausTop_{CHaus} as a reflective subcategory of all of TopTop.

The Stone-Cech compactification is in general “very large”, even for “ordinary” non-compact spaces such as the real line.

For more details see at compactum – Stone-Čech compactification

Properties

Proposition

The unit

XβX X \longrightarrow \beta X

of the compactification adjunction (βU)(\beta \dashv U) is an embedding precisely for XX a Tychonoff space.

Examples

Example

The Stone-Cech compactification of a discrete topological space is an extremally disconnected topological space. By a theorem by Gleason, these are precisely the projective objects in the category of compact Hausdorff topological spaces.

Such spaces appear for instance as connected components of w-contractible rings as objects in the pro-étale site. See (Bhatt-Scholze 13).

References

Lecture notes include

Discussion in the context of the pro-etale site is in

See also

Last revised on November 3, 2018 at 12:52:42. See the history of this page for a list of all contributions to it.