Definition:Finite Complement Topology/Uncountable

Definition

Let $S$ be an infinite set.

Let $\tau$ be the finite complement topology on $S$.

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.

1975: W.A. Sutherland: Introduction to Metric and Topological Spaces refers to the specific instance of this where $S = \R$ as the Zariski topology.

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

• Definition:Countable Finite Complement Topology

• Results about finite complement topologies can be found here.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.1$: Topological Spaces: Example $3.1.7$

The specific example presented is where $S$ is the set of real numbers $\R$.

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $19$. Finite Complement Topology on an Uncountable Space