What is "cofinality"

n. 1 (context math countable English) The least of the cardinality of the cofinal subsets of a partially ordered set. 2 (context math uncountable English) The property of being (l/en: cofinal).

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.

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.