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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.

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  • $\begingroup$ 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. $\endgroup$ – Countable Feb 6 at 23:02
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    $\begingroup$ 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. $\endgroup$ – dan_fulea Feb 6 at 23:04
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    $\begingroup$ @Countable I think the term far predates any category theory ideas about final objects and co-categories.. $\endgroup$ – Henno Brandsma Feb 6 at 23:09
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    $\begingroup$ 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." $\endgroup$ – Noah Schweber Feb 6 at 23:11
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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’.

The OED does not yet have entries for either cofinal or confinal, so I can’t say when either came into use in the mathematical sense.

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I'm not sure who came up with the term. For me it suggest that the set is "coming with you to the end", co- as in compansion, co-traveller; -final for end, the larger elements are towards the end in an order, intuitively. So whatever direction you go towards larger elements, there will always be elements from this cofinal set too..

But one should find the first usage of the term and maybe it's explained there? Not sure if Cantor used e.g.

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