Finite character

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,

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 ${\mathcal {F}}$ .
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 (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.

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.

This article incorporates material from finite character on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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Finite character

Properties

Example

See also

References