# 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**.

### 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

Let $S$ be countably infinite.

Then $\tau$ is a **finite complement topology on a countable space**, and $\struct {S, \tau}$ is a **countable finite complement space**.

### On an Uncountable Space

Let $S$ be uncountable.

Then $\tau$ is a **finite complement topology on an uncountable space**, and $\struct {S, \tau}$ is a **uncountable finite complement space**.

## Also known as

The **finite complement topology** is also called the **cofinite topology**.

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 Oscar Zariski.

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

## Also see

- Results about
**finite complement topologies**can be found**here**.

## Sources

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Chapter $\text {I}$: Topological Spaces: $1$. Open Sets and Closed Sets - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 2$: Topological Spaces: Example $6$ - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $18 \text { - } 19$. Finite Complement Topology - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**Zariski topology**