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.

It is also known as a cocountable topology or cocountable space.

On a Countable Space
It is possible to define the countable complement topology on a countable 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.

On an Uncountable Space
If $S$ is uncountable, $\tau$ is a countable complement topology on an uncountable space, and $\left({S, \tau}\right)$ is an uncountable countable complement space or uncountable cocountable space.

This distinction is rarely needed to be used, as it is normally taken for granted that $S$ is an uncountable set in the first place.

Also see

 * The Countable Complement Topology is a Topology.