Let $(A, \leq)$ be a poset. A set $B \subseteq A$ is said to be cofinal in $A$ if every $a \in A$ has a $b\in B$ with $a \leq b$.
Why is a cofinal set called "cofinal?" Is there a related notion of a "final" set? The "co" doesn't seem to be related to the word "complement," as the complement of a cofinal set can also be cofinal; such sets don't seem to be "final" in any natural sense of the word I can imagine.