Real line in the 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:

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 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 Oscar Zariski.

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

• 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 Steenand J. Arthur Seebach, Jr.: Counterexamples in Topology[2nd ed.]... [previous]... [next]: Part $\text {II}$: Counterexamples: $18 \text { - } 19$. Finite Complement Topology
• 1989:Ephraim J. Borowskiand Jonathan M. Borwein: Dictionary of Mathematics... [previous]... [next]: Entry: Zariski topology

• Definitions/Examples of Topologies
• Definitions/Finite Complement Topology