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.
Any superset of a cofinal subsets is itself cofinal.
If 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.
Subset relations and neighborhood bases
Let be a topological space and let denotes 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
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