Cocountable topology

The cocountable topology or countable complement topology on any set X consists of the empty set and all cocountable subsets of X, that is all sets whose complement in X is countable. It follows that the only closed subsets are X and the countable subsets of X.

Definition

Let $X$ be an infinite set and let ${\mathcal {T}}$ be the set of subsets of $X$ such that $$H\in {\mathcal {T}}\iff X\setminus H{\mbox{ is countable, or}}\,H=\varnothing$$ then ${\mathcal {T}}$ is the countable complement toplogy on $X$, and the topological space $T=(X,{\mathcal {T}})$ is a countable complement space.

Symbolically, the topology is typically written as $${\mathcal {T}}=\{H\subseteq X:H=\varnothing {\mbox{ or }}X\setminus H{\mbox{ is countable}}\}.$$

Proof that cocountable topology is a topology

By definition, the empty set $\varnothing$ is an element of ${\mathcal {T}}$. Similarly, the entire set $X\in {\mathcal {T}}$, since the complement of $X$ relative to itself is the empty set, which is vacuously countable.

Suppose $A,B\in {\mathcal {T}}$. Let $H=A\cap B$. Then

$X\setminus H=X\setminus (A\cap B)=(X\setminus A)\cup (X\setminus B)$

by De Morgan's laws. Since $A,B\in {\mathcal {T}}$, it follows that $X\setminus A$ and $X\setminus B$ are both countable. Because the countable union of countable sets is countable, $X\setminus H$ is also countable. Therefore, $H=A\cap B\in {\mathcal {T}}$, as its complement is countable.

Now let ${\mathcal {U}}\subseteq {\mathcal {T}}$. Then

$X\setminus \left(\bigcup {\mathcal {U}}\right)=\bigcap _{U\in {\mathcal {U}}}(X\setminus U)$

again by De Morgan's laws. For each $U\in {\mathcal {U}}$, $X\setminus U$ is countable. The countable intersection of countable sets is also countable (assuming ${\mathcal {U}}$ is countable), so $S\setminus \left(\bigcup {\mathcal {U}}\right)$ is countable. Thus, $\bigcup {\mathcal {U}}\in {\mathcal {T}}$.

Since all three open set axioms are met, ${\mathcal {T}}$ is a topology on $X$.

Properties

Every set X with the cocountable topology is Lindelöf, since every nonempty open set omits only countably many points of X. It is also T₁, as all singletons are closed.

If X is an uncountable set then any two nonempty open sets intersect, hence the space is not Hausdorff. However, in the cocountable topology all convergent sequences are eventually constant, so limits are unique. Since compact sets in X are finite subsets, all compact subsets are closed, another condition usually related to the Hausdorff separation axiom.

The cocountable topology on a countable set is the discrete topology. The cocountable topology on an uncountable set is hyperconnected, thus connected, locally connected and pseudocompact, but neither weakly countably compact nor countably metacompact, hence not compact.

See also

• Cofinite topology
• List of topologies

References

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446 (See example 20).