nLab
profinite completion of a space

(Beware there are two possible interpretations of this term. One is handled in the entry on profinite completion of a group, being profinite completion of the homotopy type of a space. The entry here treats another more purely topological concept.)

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

Idea

The profinite completion functor (on topological spaces) is the left adjoint to the inclusion of the category of profinite spaces into that of all topological spaces. It is particularly useful when applied to discrete topological spaces (i.e. really: sets!).

Definition:

Let XX be a topological space. A profinite completion of XX is a profinite space, X^\hat{X}, together with a continuous map, η X:XX^\eta_X : X \to \hat{X}, such that, if given any profinite space, YY, and a continuous map, g:XYg : X \to Y, there is a unique continuous map ψ:X^Y\psi : \hat{X}\to Y with ψη X=g\psi \eta_X = g

References

  • Abolfazl Tarizadeh, On the category of profinite spaces as a reflective subcategory (arXiv:1207.5963)

Last revised on November 20, 2013 at 12:38:08. See the history of this page for a list of all contributions to it.