In mathematics, a subset of a preordered set is said to be cofinal or frequent).

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 is referred to as the cofinality

Definitions

be a homogeneous binary relation is said to be or if it satisfies the following condition:

A subset that is not frequent is called . is a directed set, which is a preordered set with additional properties.

1. Final functions

A map between two directed sets is said to be if the image

1. Coinitial subsets

is said to be (or in the sense of forcing) if it satisfies the following condition:

This is the order-theoretic dual to the notion of cofinal subset. Cofinal (respectively coinitial) subsets are precisely the dense sets with respect to the right (respectively left) order topology.

Properties

The cofinal relation over partially ordered sets ("posetss") is reflexive: every poset is cofinal in itself. It is also transitive

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 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.

admits a totally ordered that is well-ordered

is a directed set is also a directed set.

Examples and sufficient conditions

Any superset of a cofinal subset is itself cofinal.

is cofinal. is directed.

1. Subset relations and neighborhood bases

be a topological space denote the neighborhood filter The superset is a partial order is called a .)

1. Cofinal subsets of the real numbers

but it is of natural numbers but this is

but it is but this is of all integers

Cofinal set of subsets

is a subset of the power set

be a group be the set of normal subgroups of finite index. The profinite completion is defined to be the inverse limit of the inverse system of finite quotients

See also

References