cofinite and cocountable topologies

The cofinite topology on a set $X$ is defined to be the topology $𝒯$ where

$$𝒯=\{A\subseteq X\mid X\setminus A\text{ is finite, or }A=\mathrm{\varnothing }\}.$$

In other words, the closed sets in the cofinite topology are $X$ and the finite subsets of $X$.

Analogously, the cocountable topology on $X$ is defined to be the topology in which the closed sets are $X$ and the countable subsets of $X$.

The cofinite topology on $X$ is the coarsest $T_1$ topology (http://planetmath.org/T1Space) on $X$.

The cofinite topology on a finite set $X$ is the discrete topology. Similarly, the cocountable topology on a countable set $X$ is the discrete topology.

A set $X$ together with the cofinite topology forms a compact topological space.

Title	cofinite and cocountable topologies
Canonical name	CofiniteAndCocountableTopologies
Date of creation	2013-03-22 13:03:30
Last modified on	2013-03-22 13:03:30
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	21
Author	yark (2760)
Entry type	Definition
Classification	msc 54B99
Related topic	FiniteComplementTopology
Defines	cofinite topology
Defines	cocountable topology
Defines	cofinite
Defines	cocountable