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

{{cite book

|last=Jech |first=Thomas J. |author-link=Thomas Jech

|title=The Axiom of Choice

|publisher=Dover Publications

|year=2008 |orig-year=1973

|isbn=978-0-486-46624-8}}

{{cite book

|last1=Smullyan |first1=Raymond M. |author1-link=Raymond Smullyan

|last2=Fitting |first2=Melvin |author2-link=Melvin Fitting

|title=Set Theory and the Continuum Problem

|publisher=Dover Publications

|year=2010 |orig-year=1996

|isbn=978-0-486-47484-7}}

Category:Families of sets

finite character

Properties

Example

See also

References