Finite character

In mathematics, a family ${ℱ}$ of sets is of finite character if for each $A$, $A$ belongs to ${ℱ}$ if and only if every finite subset of $A$ belongs to ${ℱ}$. That is,

1. For each $A\in {ℱ}$, every finite subset of $A$ belongs to ${ℱ}$.
2. If every finite subset of a given set $A$ belongs to ${ℱ}$, then $A$ belongs to ${ℱ}$.

Properties

A family ${ℱ}$ of sets of finite character enjoys the following properties:

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

Example

Let $V$ be a vector space, and let ${ℱ}$ be the family of linearly independent subsets of $V$. Then ${ℱ}$ is a family of finite character (because a subset $X\subseteq V$ is linearly dependent if and only if $X$ 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.