Definition:Finite Character

Definition

Let $S$ be a set.

Let $A$ be a set of subsets of $S$.

Then $A$ has finite character if and only if for each $x \subseteq S$:

$x \in A$ if and only if every finite subset of $x$ is in $A$.

Property of Sets

Let $P$ be a property of sets.

Then $P$ has finite character if and only if for every set $x$:

$x$ has property $P$ if and only if every finite subset of $x$ has property $P$.

Class Theory

In the context of class theory, the definition follows the same lines:

Let $A$ be a class.

Then $A$ has finite character if and only if, for all $x$:

$x \in A$ if and only if every finite subset of $x$ is in $A$.

Also known as

To say that:

$A$ has finite character

is the same as saying that:

$A$ is of finite character.

Also see

• Tukey's Lemma, an equivalent of the Axiom of Choice
• Restricted Tukey's Theorem, an equivalent of the Boolean Prime Ideal Theorem.

• Results about finite character can be found here.

Sources

• 2005: R.E. Hodel: Restricted versions of the Tukey-Teichmuller Theorem that are equivalent to the Boolean prime ideal theorem (Arch. Math. Logic Vol. 44: pp. 459 – 472)