What is "cofinal"

a. (context order theory English) Of a subset of a (l/en: partially ordered set); containing elements at least as late as any given element of the set, relative to the given partial order.

Cofinal may refer to:

• Cofinal (mathematics)
• Cofinality (mathematics)
• Cofinal (music), a part of some Gregorian chants

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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 ≤.

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.