# cofinite and cocountable topologies

The cofinite topology  on a set $X$ is defined to be the topology  $\mathcal{T}$ where

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

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 CofiniteAndCocountableTopologies 2013-03-22 13:03:30 2013-03-22 13:03:30 yark (2760) yark (2760) 21 yark (2760) Definition msc 54B99 FiniteComplementTopology cofinite topology cocountable topology cofinite cocountable