Finite character

In mathematics, a family of sets is of finite character if for each , belongs to if and only if every finite subset of belongs to . That is,

  1. For each , every finite subset of belongs to .
  2. 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:

  1. For each , every (finite or infinite) subset of belongs to .
  2. Every nonempty family of finite character has a maximal element with respect to inclusion (Tukey's lemma): In , partially ordered by inclusion, the union of every chain of elements of also belongs to , therefore, by Zorn's lemma, contains at least one maximal element.

Example

Let be a vector space, and let be the family of linearly independent subsets of . Then is a family of finite character (because a subset is linearly dependent if and only if 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

References

  • Jech, Thomas J. (2008) [1973]. The Axiom of Choice. Dover Publications. ISBN 978-0-486-46624-8.
  • Smullyan, Raymond M.; Fitting, Melvin (2010) [1996]. Set Theory and the Continuum Problem. Dover Publications. ISBN 978-0-486-47484-7.

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