Set of Subsets of Finite Character of Countable Set is of Type M

Theorem

Let $D$ be a countable set.

Let $S$ be a set of subsets of $D$ such that $S$ is of finite character.

Then $S$ is of type $M$, that is:

every element of $S$ is a subset of a maximal element of $S$ under the subset relation.

Proof

We have that $\ds \bigcup S \subseteq D$.

Hence by Subset of Countable Set is Countable, $\ds \bigcup S$ is itself countable.

Hence from Countable Set has Choice Function, $S$ has a choice function.

The result follows from Set of Finite Character with Choice Function is of Type $M$.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text {II}$ -- Maximal principles: $\S 5$ Maximal principles