# Definition:Countable Complement Topology

## Definition

Let $S$ be an infinite set whose cardinality is usually taken to be uncountable.

Let $\tau$ be the set of subsets of $S$ defined as:

- $H \in \tau \iff \complement_S \left({H}\right)$ is countable, or $H = \varnothing$

where $\complement_S \left({H}\right)$ denotes the complement of $H$ relative to $S$.

In this definition, countable is used in its meaning that includes finite.

Then $\tau$ is the **countable complement topology on $S$**, and the topological space $T = \left({S, \tau}\right)$ is a **countable complement space**.

### On a Countable Space

It is possible to define the countable complement topology on a countable set set $S$, but as every subset of a countable set has a countable complement, it is clear that this is trivially equal to the discrete space.

This is why the countable complement topology is usually understood to apply to uncountable sets only.

## Also known as

A **countable complement topology** is also known as a **co-countable topology**, and similarly we have the term **co-countable space**.

Sometimes the hyphen is omitted, to give the word **cocountable**.

## Also see

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

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 20$