nLab
cofinite topology

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

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

  1. all cofinite subsets;

  2. the empty set.

Properties

If XX is a finite set, then its cofinite topology coincides with its discrete topology.

The cofinite topology on a set XX is the coarsest topology on XX that satisfies the T 1T_1 separation axiom, hence the condition that every singleton subset is a closed subspace.

Indeed, every T 1T_1-topology on XX has to be finer that the cofinite topology.

If XX 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 1T_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. See the history of this page for a list of all contributions to it.