In mathematics, a subset of a preordered set is said to be cofinal or frequent in if for every it is possible to find an element in that is "larger than " (explicitly, "larger than " means ).
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 is referred to as the cofinality of
Let be a homogeneous binary relation on a set
A subset is said to be cofinal or frequent with respect to if it satisfies the following condition:
- For every there exists some that
A subset that is not frequent is called infrequent.
This definition is most commonly applied when is a directed set, which is a preordered set with additional properties.
- Final functions
A map between two directed sets is said to be final if the image of is a cofinal subset of
- Coinitial subsets
A subset is said to be coinitial (or dense in the sense of forcing) if it satisfies the following condition:
- For every there exists some such that
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. It is also transitive: if is a cofinal subset of a poset and is a cofinal subset of (with the partial ordering of applied to ), then is also a cofinal subset of
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 admits a totally ordered cofinal subset, then we can find a subset that is well-ordered and cofinal in
If is a directed set and if is a cofinal subset of then is also a directed set.
Examples and sufficient conditions
Any superset of a cofinal subset is itself cofinal.
If is a directed set and if some union of (one or more) finitely many subsets is cofinal then at least one of the set is cofinal. This property is not true in general without the hypothesis that is directed.
- Subset relations and neighborhood bases
Let be a topological space and let denote the neighborhood filter at a point
The superset relation is a partial order on : explicitly, for any sets and declare that if and only if (so in essence, is equal to ).
A subset is called a neighborhood base at if (and only if) is a cofinal subset of
that is, if and only if for every there exists some such that (i.e. such that ).
- Cofinal subsets of the real numbers
For any the interval is a cofinal subset of but it is not a cofinal subset of
The set of natural numbers (consisting of positive integers) is a cofinal subset of but this is not true of the set of negative integers
Similarly, for any the interval is a cofinal subset of but it is not a cofinal subset of
The set of negative integers is a cofinal subset of but this is not true of the natural numbers
The set of all integers is a cofinal subset of and also a cofinal subset of ; the same is true of the set
Cofinal set of subsets
A particular but important case is given if is a subset of the power set of some set ordered by reverse inclusion Given this ordering of a subset is cofinal in if for every there is a such that
For example, let be a group and let be the set of normal subgroups of finite index. The profinite completion of is defined to be the inverse limit of the inverse system of finite quotients of (which are parametrized by the set ).
In this situation, every cofinal subset of is sufficient to construct and describe the profinite completion of
- Cofinality – Size of subsets in order theory
- Upper set – Subset of a preorder that contains all larger elements
- a subset of a partially ordered set that contains every element for which there is an with