Set of Finite Character with Countable Union is Type M
Theorem
Let $S$ be a set of sets of finite character.
Let its union $\bigcup S$ be countable.
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 1
By Countable Set has Choice Function, $S$ has a choice function.
The result follows from Set of Finite Character with Choice Function is Type $M$.
$\blacksquare$
Proof 2
Let $S$ be a set of sets of finite character whose union $\bigcup S$ is countable.
Let $D := \bigcup S$ be enumerated as:
- $D = \set {d_1, d_2, \ldots, d_n, d_{n + 1}, \ldots}$
Let $b \in S$ be arbitrary.
From the Principle of Recursive Definition, we can generate a countable sequence of elements of $S$ as follows:
Let $b_0 := b$.
Let it be assumed that $b_n$ has been defined.
Then we take:
- $b_{n + 1} := \begin{cases} b_n \cup \set {d_n} & : \text {if $b_n \cup \set {d_n}$ is an element of $S$} \\ b_n & : \text {otherwise} \end{cases}$
Let $C$ be the set defined as
- $C := \set {b_0, b_1, \ldots, b_n, b_{n + 1}, \ldots}$
From Increasing Sequence of Sets forms Nest, $C$ is a nest which, as $C$ is a set, is in fact a chain.
Hence $\bigcup C \in S$.
From Union of Chain in Set of Finite Character with Countable Union is Maximal Element, $\bigcup C$ is a maximal element of $S$ under the subset relation.
Thus $b$ is a subset of a maximal element of $S$ under the subset relation.
As $b$ is arbitrary, the result follows.
$\blacksquare$