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

open subset , closed subset , neighbourhood

topological space , locale

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

uniformly continuous function

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

Universal constructions

Extra stuff, structure, properties

nice topological space

metric space , metric topology , metrisable space

Kolmogorov space , Hausdorff space , regular space , normal space

sober space

compact space , proper map

sequentially compact , countably compact , locally compact , sigma-compact , paracompact , countably paracompact , strongly compact

compactly generated space

second-countable space , first-countable space

contractible space , locally contractible space

connected space , locally connected space

simply-connected space , locally simply-connected space

cell complex , CW-complex

pointed space

topological vector space , Banach space , Hilbert space

topological group

topological vector bundle , topological K-theory

topological manifold

Examples

empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

cylinder , cone

sphere , ball

circle , torus , annulus , Moebius strip

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

path , loop

mapping spaces : compact-open topology , topology of uniform convergence

Zariski topology

Cantor space , Mandelbrot space

Peano curve

line with two origins , long line , Sorgenfrey line

K-topology , Dowker space

Warsaw circle , Hawaiian earring space

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 $X$ be a topological space . A profinite completion of $X$ is a profinite space , $\hat{X}$ , together with a continuous map , $\eta_X : X \to \hat{X}$ , such that, if given any profinite space, $Y$ , and a continuous map, $g : X \to Y$ , there is a unique continuous map $\psi : \hat{X}\to Y$ with $\psi \eta_X = g$

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.
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