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:
 * $H \in \tau \iff \relcomp S H$ is countable, or $H = \O$

where $\relcomp S H$ 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 $T = \struct {S, \tau}$ is a countable complement space.

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

 * Countable Complement Topology is Topology