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 subset of belongs to .
  2. If we take the axiom of choice to be true then every nonempty family of finite character has a maximal element with respect to inclusion : 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.
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 vector basis.