# Why is a cofinal set called “cofinal?”

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.

• I think it should mean that in the right category (which seems to be a subcategory of $\textbf{Pos}$ which somehow depends on A), there is only one morphism (order preserving map with possibly some other restrictions depending on what this new category might be) from B to every other object in that same category. – Countable Feb 6 at 23:02
• The co prefix in the word cofinal has for me the touch of "together with" as in Maths & CO. Well, $A$ and its cofinal set $B$ have the same "chance" to inspect the final, terminal properties of the order. – dan_fulea Feb 6 at 23:04
• @Countable I think the term far predates any category theory ideas about final objects and co-categories.. – Henno Brandsma Feb 6 at 23:09
• FWIW there is a sort of "double-negation duality" going on here: $B$ is cofinal in $A$ iff $A\setminus B$ is not a "final segment" of $A$ (although the latter term is really most used in the context of linear orders). So "cofinal" = "complement not final." – Noah Schweber Feb 6 at 23:11

The term cofinal is a compound of the prefix co- and the adjective final. Here the meaning of the prefix is ‘together; mutually; jointly’: if $$B$$ is cofinal in $$A$$, the two sets reach the ‘end’ together, so to speak. The prefix is from Latin co-, a variant of con-, and indeed one occasionally sees confinal in place of cofinal. The Latin prefix con- in turn is derived from the preposition cum ‘with, along with’.