Finite character

In mathematics, a family ${\displaystyle {\mathcal {F}}}$ of sets is of finite character if for each ${\displaystyle A}$, ${\displaystyle A}$ belongs to ${\displaystyle {\mathcal {F}}}$ if and only if every finite subset of ${\displaystyle A}$ belongs to ${\displaystyle {\mathcal {F}}}$. That is,

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

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

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

Let ${\displaystyle V}$ be a vector space, and let ${\displaystyle {\mathcal {F}}}$ be the family of linearly independent subsets of ${\displaystyle V}$. Then ${\displaystyle {\mathcal {F}}}$ is a family of finite character (because a subset ${\displaystyle X\subseteq V}$ is linearly dependent if and only if ${\displaystyle X}$ 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.

• Hereditarily finite set

• 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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