Finite character

Not to be confused with Character of a finite group.

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

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

Properties

A family $ℱ$ of sets of finite character enjoys the following properties:

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

Example

Let $V$ be a vector space, and let $ℱ$ be the family of linearly independent subsets of $V$. Then $ℱ$ 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

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