nLab
cofinite topology
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

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
Definition
Given a set $X$ , then the cofinite topology or finite complement topology on $X$ is the topology whose open subsets are precisely

all cofinite subsets ;

the empty set .

Properties
If $X$ is a finite set , then its cofinite topology coincides with its discrete topology .

The cofinite topology on a set $X$ is the coarsest topology on $X$ that satisfies the $T_1$ separation axiom , hence the condition that every singleton subset is a closed subspace .

Indeed, every $T_1$ -topology on $X$ has to be finer that the cofinite topology.

If $X$ is not finite , then its cofinite topology is not sober , hence in particular not Hausdorff (since Hausdorff implies sober ).

A set equipped with the cofinite topology forms a compact space . However, this type of compact space is not uniformizable ; if it were, then under the $T_1$ condition it would also be Hausdorff, which as we saw is not the case.

References
Last revised on June 3, 2017 at 10:59:20.
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