# Definition:Countable Complement Topology

## Definition

Let $S$ be an infinite set whose cardinality is usually taken to be uncountable.

Let $\tau$ be the set of subsets of $S$ defined as:

$H \in \tau \iff \complement_S \left({H}\right)$ is countable, or $H = \varnothing$

where $\complement_S \left({H}\right)$ denotes the complement of $H$ relative to $S$.

In this definition, countable is used in its meaning that includes finite.

Then $\tau$ is the countable complement topology on $S$, and the topological space $T = \left({S, \tau}\right)$ is a countable complement space.

### On a Countable Space

It is possible to define the countable complement topology on a countable set set $S$, but as every subset of a countable set has a countable complement, it is clear that this is trivially equal to the discrete space.

This is why the countable complement topology is usually understood to apply to uncountable sets only.

## Also known as

A countable complement topology is also known as a co-countable topology, and similarly we have the term co-countable space.

Sometimes the hyphen is omitted, to give the word cocountable.

## Also see

• Results about countable complement topologies can be found here.