Finite character

In mathematics, a family ${\displaystyle {\mathcal {F}}}$ of sets is of finite character provided it has the following properties:

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

Properties

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

1. For each ${\displaystyle A\in {\mathcal {F}}}$, every (finite or infinite) subset of ${\displaystyle A}$ belongs to ${\displaystyle {\mathcal {F}}}$.
2. Tukey's lemma: In ${\displaystyle {\mathcal {F}}}$, partially ordered by inclusion, the union of every chain of elements of ${\displaystyle {\mathcal {F}}}$ also belong to ${\displaystyle {\mathcal {F}}}$, therefore, by Zorn's lemma, ${\displaystyle {\mathcal {F}}}$ 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 XV is linearly dependent iff 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.