Cofinal subset

Summary

A subset $C$ of a preordered set $P$ is cofinal when every element of $P$ lies below some element of $C$. A cofinal subset reaches arbitrarily far up in $P$. For example, $C$ and $P$ have the same upper bounds, so a supremum of $P$ can be computed from $C$ alone. The least cardinality of a cofinal subset is the cofinality of $P$.