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

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

Example

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