# Definition:Finite Complement Topology/Uncountable

< Definition:Finite Complement Topology(Redirected from Definition:Uncountable Finite Complement Topology)

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## 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

- 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