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:
Contents
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.
Properties
The cofinal relation over partially ordered sets ("poset") is reflexive: every poset is cofinal in itself. It is also transitive: if B is a cofinal subset of a poset A, and C is a cofinal subset of B (with the partial ordering of A applied to B), then C is also a cofinal subset of A.
For a partially ordered set with maximal elements, every cofinal subset must contain all maximal elements, otherwise a maximal element which is not in the subset would fail to be less than any element of the subset, violating the definition of cofinal. For a partially ordered set with a greatest element, a subset is cofinal if and only if it contains that greatest element (this follows, since a greatest element is necessarily a maximal element). Partially ordered sets without greatest element or maximal elements admit disjoint cofinal subsets. For example, the even and odd natural numbers form disjoint cofinal subsets of the set of all natural numbers.
If a partially ordered set A admits a totally ordered cofinal subset, then we can find a subset B which is well-ordered and cofinal in A.
Cofinal set of subsets
A particular but important case is given if A is a subset of the power set P(E) of some set E, ordered by reverse inclusion (⊃). Given this ordering of A, a subset B of A is cofinal in A if for every a ∈ A there is a b ∈ B such that a ⊃ b.
For example, if E is a group, A could be the set of normal subgroups of finite index. Then, cofinal subsets of A (or sequences, or nets) are used to define Cauchy sequences and the completion of the group.
Related Notions
A map ƒ: X → A between two directed sets is said to be final if the range ƒ(X) of f is a cofinal subset of A.