- For each , every finite subset of belongs to .
- If every finite subset of a given set belongs to , then belongs to .
A family of sets of finite character enjoys the following properties:
- For each , every (finite or infinite) subset of belongs to .
- Tukey's lemma: In , partially ordered by inclusion, the union of every chain of elements of also belong to , therefore, by Zorn's lemma, contains at least one maximal element.
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 ⊆ V 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.
|This mathematical logic-related article is a stub. You can help Wikipedia by expanding it.|