Cofinal (mathematics)

Cofinal (mathematics)

In mathematics, let A be a set and let ≤ be a binary relation on A. Then a subset B of A is said to be cofinal if it satisfies the following condition:

For every a ∈ A, there exists some b ∈ B such that a ≤ b.

This definition is most commonly applied when A is a partially ordered set or directed set under the relation ≤. Also, the notion of cofinal is sometimes applied to objects other than subsets, e.g. a cofinal function ƒ: X → A is a function whose range ƒ(X) is a cofinal subset of A

Cofinal subsets are very important in the theory of directed sets and nets, where “cofinal subnet” is the appropriate generalization of “subsequence”. They are also important in order theory, including the theory of cardinal numbers, where the minimum possible cardinality of a cofinal subset of A is referred to as the cofinality of A.

A subset B of A is said to be coinitial (or dense in the sense of forcing) if it satisfies the following condition:

For every a ∈ A, there exists some b ∈ B such that b ≤ a.

This is the order-theoretic dual to the notion of cofinal subset.

Note that cofinal and coinitial subsets are both dense in the sense of appropriate (right- or left-) order topology.