Finite character


In mathematics, a family of sets is of finite character provided it has the following properties:
  1. For each, every finite subset of belongs to .
  2. If every finite subset of a given set belongs to , then belongs to .

    Properties

A family of sets of finite character enjoys the following properties:
  1. For each, every subset of belongs to .
  2. 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.

    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.
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 vector basis.