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{{Distinguish|Character of a finite group}}In mathematics, a family mathcal{F} of sets is of finite character if for each A, A belongs to mathcal{F} if and only if every finite subset of A belongs to mathcal{F}. That is,

1. For each Ain mathcal{F}, every finite subset of A belongs to mathcal{F}.
2. 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:

1. For each Ain mathcal{F}, every (finite or infinite) subset of A belongs to mathcal{F}.
2. Every nonempty family of finite character has a maximal element with respect to inclusion (Tukey's lemma): In mathcal{F}, partially ordered by inclusion, the union of every chain of elements of mathcal{F} also belongs 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 mathcal{F} be the family of linearly independent subsets of V. Then mathcal{F} is a family of finite character (because a subset X subseteq V is linearly dependent if and only if 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.

See also

• Hereditarily finite set

References

• BOOK

• BOOK

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