Editing Cofinality (section)

==Examples==

*  The cofinality of a partially ordered set with [[greatest element]] is 1 as the set consisting only of the greatest element is cofinal (and must be contained in every other cofinal subset).
** In particular, the cofinality of any nonzero finite ordinal, or indeed any finite directed set, is 1, since such sets have a greatest element.
* Every cofinal subset of a partially ordered set must contain all [[maximal element]]s of that set. Thus the cofinality of a finite partially ordered set is equal to the number of its maximal elements.
** In particular, let <math>A</math> be a set of size <math>n,</math> and consider the set of subsets of <math>A</math> containing no more than <math>m</math> elements.  This is partially ordered under inclusion and the subsets with <math>m</math> elements are maximal.  Thus the cofinality of this poset is <math>n</math> [[Binomial coefficient|choose]] <math>m.</math>
* A subset of the [[natural number]]s <math>\N</math> is cofinal in <math>\N</math> if and only if it is infinite, and therefore the cofinality of <math>\aleph_0</math> is <math>\aleph_0.</math> Thus <math>\aleph_0</math> is a [[regular cardinal]].
* The cofinality of the [[real number]]s with their usual ordering is <math>\aleph_0,</math> since <math>\N</math> is cofinal in <math>\R.</math>  The usual ordering of <math>\R</math> is not [[order isomorphic]] to <math>c,</math> the [[Cardinality of the continuum|cardinality of the real numbers]], which has cofinality strictly greater than <math>\aleph_0.</math>  This demonstrates that the cofinality depends on the order; different orders on the same set may have different cofinality.