Finite character

In mathematics, a family $$\mathcal{F}$$ of sets is of finite character if for each $$A$$, $$A$$ belongs to  $$\mathcal{F}$$ if and only if every finite subset of $$A$$ belongs to  $$\mathcal{F}$$. That is,
 * 1) For each $$A\in \mathcal{F}$$, every finite subset of $$A$$ belongs to $$\mathcal{F}$$.
 * 2) If every finite subset of a given set $$A$$ belongs to $$\mathcal{F}$$, then $$A$$ belongs to  $$\mathcal{F}$$.

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


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

Example
Let $$V$$ be a vector space, and let $$\mathcal{F}$$ be the family of linearly independent subsets of $$V$$. Then $$\mathcal{F}$$ 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.