Superset of Co-Countable Set
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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.
$\blacksquare$