In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A. Formally,

This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member. The cofinality of a partially ordered set A can alternatively be defined as the least ordinal x such that there is a function from x to A with cofinal image. This second definition makes sense without the axiom of choice. If the axiom of choice is assumed, as will be the case in the rest of this article, then the two definitions are equivalent.

Cofinality can be similarly defined for a directed set and is used to generalize the notion of a subsequence in a net.

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. The cofinality of any successor ordinal is 1.

Every cofinal subset of a partially ordered set must contain all maximal elements of that set. Thus the cofinality of a finite partially ordered set is equal to the number of its maximal elements.

• choose

• A subset of the natural numbers is a regular cardinal.

• The cofinality of the real numbers is not order isomorphic the cardinality of the real numbers This demonstrates that the cofinality depends on the order; different orders on the same set may have different cofinality.

Properties

admits a totally ordered that is well-ordered with minimal order type will be order isomorphic.

Cofinality of ordinals and other well-ordered sets

The cofinality of an ordinal that is the order type of a cofinal subset The cofinality of a set of ordinals or any other well-ordered set is the cofinality of the order type of that set.

Thus for a limit ordinal or an uncountable cofinality.

The cofinality of 0 is 0. The cofinality of any successor ordinal is 1. The cofinality of any nonzero limit ordinal is an infinite regular cardinal.

Regular and singular ordinals

A regular ordinal is an ordinal that is equal to its cofinality. A singular ordinal is any ordinal that is not regular.

Every regular ordinal is the initial ordinal are initial ordinals that are not regular.

So the cofinality operation is idempotent.

Cofinality of cardinals

is the least cardinal such that there is an unbounded

that is, the disjoint union

Using Kőnig's theorem

(Compare to the continuum hypothesis)

On the other hand, if the axiom of choice

See also

References

• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. .

• Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. .