Finite character

In mathematics, a family ${\mathcal {F}}$ of sets is of finite character provided it has the following properties:

For each $A\in {\mathcal {F}}$ , every finite subset of $A$ belongs to ${\mathcal {F}}$ .
If every finite subset of a given set $A$ belongs to ${\mathcal {F}}$ , then $A$ belongs to ${\mathcal {F}}$ .

Properties

A family ${\mathcal {F}}$ of sets of finite character enjoys the following properties:

For each $A\in {\mathcal {F}}$ , every (finite or infinite) subset of $A$ belongs to ${\mathcal {F}}$ .
Tukey's lemma: In ${\mathcal {F}}$ , partially ordered by inclusion, the union of every chain of elements of ${\mathcal {F}}$ also belong to ${\mathcal {F}}$ , therefore, by Zorn's lemma, ${\mathcal {F}}$ 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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