Finite character

In mathematics, a family [math]\displaystyle{ \mathcal{F} }[/math] of sets is of finite character if for each [math]\displaystyle{ A }[/math], [math]\displaystyle{ A }[/math] belongs to [math]\displaystyle{ \mathcal{F} }[/math] if and only if every finite subset of [math]\displaystyle{ A }[/math] belongs to [math]\displaystyle{ \mathcal{F} }[/math]. That is,

1. For each [math]\displaystyle{ A\in \mathcal{F} }[/math], every finite subset of [math]\displaystyle{ A }[/math] belongs to [math]\displaystyle{ \mathcal{F} }[/math].
2. If every finite subset of a given set [math]\displaystyle{ A }[/math] belongs to [math]\displaystyle{ \mathcal{F} }[/math], then [math]\displaystyle{ A }[/math] belongs to [math]\displaystyle{ \mathcal{F} }[/math].

Properties

A family [math]\displaystyle{ \mathcal{F} }[/math] of sets of finite character enjoys the following properties:

1. For each [math]\displaystyle{ A\in \mathcal{F} }[/math], every (finite or infinite) subset of [math]\displaystyle{ A }[/math] belongs to [math]\displaystyle{ \mathcal{F} }[/math].
2. Every nonempty family of finite character has a maximal element with respect to inclusion (Tukey's lemma): In [math]\displaystyle{ \mathcal{F} }[/math], partially ordered by inclusion, the union of every chain of elements of [math]\displaystyle{ \mathcal{F} }[/math] also belongs to [math]\displaystyle{ \mathcal{F} }[/math], therefore, by Zorn's lemma, [math]\displaystyle{ \mathcal{F} }[/math] contains at least one maximal element.

Example

Let [math]\displaystyle{ V }[/math] be a vector space, and let [math]\displaystyle{ \mathcal{F} }[/math] be the family of linearly independent subsets of [math]\displaystyle{ V }[/math]. Then [math]\displaystyle{ \mathcal{F} }[/math] is a family of finite character (because a subset [math]\displaystyle{ X \subseteq V }[/math] is linearly dependent if and only if [math]\displaystyle{ X }[/math] has a finite subset which is linearly dependent). Therefore, in every vector space, there exists a maximal family of linearly independent elements. As a maximal family is a vector basis, every vector space has a (possibly infinite) vector basis.

See also

• Hereditarily finite set

References

• The Axiom of Choice. Dover Publications. 2008. ISBN 978-0-486-46624-8. 
• Set Theory and the Continuum Problem. Dover Publications. 2010. ISBN 978-0-486-47484-7.