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
be a homogeneous binary relation on a set
is said to be cofinal or frequent[1] with respect to
if it satisfies the following condition: A subset that is not frequent is called infrequent.
[1] This definition is most commonly applied when
between two directed sets is said to be final[2] if the image
is said to be coinitial (or dense 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.
The cofinal relation over partially ordered sets ("posets") is reflexive: every poset is cofinal in itself.
is a cofinal subset of a poset
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.
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
is a directed set and if some union of (one or more) finitely many subsets
[1] This property is not true in general without the hypothesis that
denote the neighborhood filter at a point
is called a neighborhood base at
of natural numbers (consisting of positive integers) is a cofinal subset of
but this is not true of the set of negative integers
of negative integers is a cofinal subset of
but this is not true of the natural numbers
of all integers is a cofinal subset of
is a subset of the power set
ordered by reverse inclusion
be the set of normal subgroups of finite index.
is defined to be the inverse limit of the inverse system of finite quotients of
In this situation, every cofinal subset of
is sufficient to construct and describe the profinite completion of