Superset of Co-Countable Set

Theorem

Every superset of a co-countable set is co-countable.

Proof

Let $S$ be a set.

Let $A$ be co-countable in $S$, and let $B$ be such that $A \subseteq B \subseteq S$.

From Relative Complement inverts Subsets, it follows that:

$\complement_S \left({B}\right) \subseteq \complement_S \left({A}\right)$

As $A$ is co-countable, $\complement_S \left({A}\right)$ is countable.

By Subset of Countably Infinite Set is Countable, it follows that $\complement_S \left({B}\right)$ is also countable.

Therefore, $B$ is also co-countable, and the result follows.

$\blacksquare$

Also see

• Subset of Countably Infinite Set is Countable