Cofinal (mathematics)

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 is "larger than ${\displaystyle a}$" (explicitly, "larger than ${\displaystyle a}$" means ${\displaystyle a\leq b}$).

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 ${\displaystyle A.}$