Are Monotone classes always closed under complementation? I attempted to construct a counterexample however I was not able to do so.
Recall the definition of a monotone class is:
$C \subset P(X)$ for some set $X$ is called a monotone class if it is closed under countable union of an increasing sequence of sets. And similarly, is closed under countable intersection for a decreasing sequence of sets.
Best Answer
No, consider $C = \{ c \subseteq \mathbb N \mid 0 \in c \}$. $C \subseteq \mathcal P(\mathbb{N})$ is closed under unions and intersections (hence monotone) but not under complements, since $\mathbb N \setminus \{0\} \not \in C$.