Finite character
In mathematics, a family of sets is of finite character provided it has the following properties:
- For each
, every finite subset of
belongs to
.
- 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:
- 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.
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 (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.
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