# Definition:Finite Complement Topology/Uncountable

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

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## Contents

## 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**.

## Also see

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

## Sources

- 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