finite character

In mathematics, a family $\mathcal$ of sets is of finite character if for each $A$, $A$ belongs to $\mathcal$ if and only if every finite subset of $A$ belongs to $\mathcal$. That is,
#For each $A\in \mathcal$, every finite subset of $A$ belongs to $\mathcal$.
#If every finite subset of a given set $A$ belongs to $\mathcal$, then $A$ belongs to $\mathcal$.

Properties

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

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

Example

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

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{{Mathlogic-stub
Families of sets