In mathematics, a subset ${\displaystyle B\subseteq A}$ of a preordered set ${\displaystyle (A,\leq )}$ is said to be cofinal or frequent in ${\displaystyle A}$ if for every ${\displaystyle a\in A,}$ it is possible to find an element ${\displaystyle b}$ in ${\displaystyle B}$ that dominates ${\displaystyle a}$ (formally, ${\displaystyle a\leq b}$).

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 ${\displaystyle A}$ is referred to as the cofinality of ${\displaystyle A.}$

Let ${\displaystyle \,\leq \,}$ be a homogeneous binary relation on a set ${\displaystyle A.}$ A subset ${\displaystyle B\subseteq A}$ is said to be cofinal or frequent with respect to ${\displaystyle \,\leq \,}$ if it satisfies the following condition:

A subset that is not frequent is called infrequent. This definition is most commonly applied when ${\displaystyle (A,\leq )}$ is a directed set, which is a preordered set with additional properties.

A map ${\displaystyle f:X\to A}$ between two directed sets is said to be final if the image ${\displaystyle f(X)}$ of ${\displaystyle f}$ is a cofinal subset of ${\displaystyle A.}$

A subset ${\displaystyle B\subseteq A}$ 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. It is also transitive: if ${\displaystyle B}$ is a cofinal subset of a poset ${\displaystyle A,}$ and ${\displaystyle C}$ is a cofinal subset of ${\displaystyle B}$ (with the partial ordering of ${\displaystyle A}$ applied to ${\displaystyle B}$), then ${\displaystyle C}$ is also a cofinal subset of ${\displaystyle A.}$