Finite character explained
of
sets is of
finite character if for each
,
belongs to
if and only if every
finite subset of
belongs to
. That is,
- For each
, every
finite subset of
belongs to
.
- If every finite subset of a given set
belongs to
, then
belongs to
.
Properties
A family
of sets of finite character enjoys the following properties:
- For each
, every (finite or infinite)
subset of
belongs to
.
- 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
,
partially ordered by inclusion, the
union of every chain of elements of
also belongs to
, therefore, by Zorn's lemma,
contains at least one maximal element.
Example
Let
be a
vector space, and let
be the family of
linearly independent subsets of
. Then
is a family of finite character (because a subset
is linearly dependent if and only if
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
References
- Book: Jech, Thomas J. . Thomas Jech . The Axiom of Choice . . 2008 . 1973 . 978-0-486-46624-8.
- Book: Smullyan . Raymond M. . Raymond Smullyan . Fitting . Melvin . Melvin Fitting . Set Theory and the Continuum Problem . Dover Publications . 2010 . 1996 . 978-0-486-47484-7.
