Cofinal (mathematics)

In mathematics, let A be a set and let ≤ be a binary relation on A. Then a subset B ⊆ A is said to be cofinal or frequent[1] in A if it satisfies the following condition:

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

A subset that is not frequent is called infrequent.[1] 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 ⊆ 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 ∈ Bsuch 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 ("posets") 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 that is not in the subset would fail to be less than or equal to 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 that is well-ordered and cofinal in A.

If (A, ≤) is a directed set and if B ⊆ A is a cofinal subset of A then (B, ≤) is also a directed set.[1]

Examples and sufficient conditions

Any superset of a cofinal subsets is itself cofinal.[1] If (A, ≤) is a preordered set and if some union of (one or more) finitely many subsets is cofinal then at least one of the set is cofinal.[1]

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 ⊆ A is cofinal in A if for every a ∈ A there is a b ∈ Bsuch that a ⊇ b.

For example, let E be a group and let A be the set of normal subgroups of finite index. The profinite completion of E is defined to be the inverse limit of the inverse system of finite quotients of E (which are parametrized by the set A). In this situation, every cofinal subset of A is sufficient to construct and describe the profinite completion of E.

A map f : X → A between two directed sets is said to be final[2] if the range f(X) of f is a cofinal subset of A.

See also
  • Cofinite
  • Cofinality
  • Upper set – a subset U of a partially ordered set (P,≤) that contains every element y of P for which there is an x in U with x ≤ y

References
  1. Schechter 1996, pp. 158-165.
  2. Bredon, Glen (1993). Topology and Geometry. Springer. p. 16.
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