Definition:Finite Complement Topology

Definition
Let $S$ be a set whose cardinality is usually specified as being infinite.

Let $\tau$ be the set of subsets of $S$ defined as:
 * $H \in \tau \iff \complement_S \left({H}\right) \text { is finite, or } H = \varnothing$

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

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

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

This is why the finite complement topology is usually understood to apply to infinite sets only.

On a Countable Space
If $S$ is countably infinite, $\tau$ is a finite complement topology on a countable space, and $\left({S, \tau}\right)$ is a countable finite complement space.

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

Also known as
The term cofinite is sometimes seen in place of finite complement.

Some sources are more explicit about the nature of this topology, and call it the topology of finite complements.

Also see

 * Finite Complement Topology is Topology