Finite character

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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[edit]

A family of sets of finite character enjoys the following properties:

  1. For each , every (finite or infinite) 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[edit]

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.

This article incorporates material from finite character on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.