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:
 * $H \in \tau \iff \relcomp S H \text { is finite, or } H = \O$

where $\relcomp S H$ denotes the complement of $H$ relative to $S$.

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

The finite complement topology can also be referred to as the minimal $T_1$ topology (on a given set).

This is justified by Finite Complement Topology is Minimal $T_1$ Topology.

This topology is also given by some sources as the Zariski topology, for.

However, this is not recommended as there is another so named Zariski topology which is unrelated to this one.

Also see

 * Finite Complement Topology is Topology