Finite character

Properties

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

1. For each A ∈ F , every (finite or infinite) subset of A belongs to F .
2. Tukey's lemma: In F , partially ordered by inclusion, the union of every chain of elements of F also belong to F , therefore, by Zorn's lemma, F 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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