I was reading about ordered fields $\mathbb{F}$ with countable cofinality which means that there is a countable set $S$ that is cofinal. The definition of $S$ being cofinal is that for all $a \in \mathbb{F}$ there exists $s \in S$ such that $s \geq a$.
I was wondering about the meaning of the name "cofinal". It seems (at least to me) that it would imply that there is such a notion as a "final" subset. Oddly enough, the dual notion seems to be that of a coinitial subset (just switch the direction of the inequality in cofinality).
I was just wondering if anyone knew if there was any meaning to the "co-". I looked around on Wikipedia and did not really find anything. The closed that I came was the definition of an initial/lower set. So, I suppose that to dualize the definition of initial set, you take its definition: $\forall s \in S, (y \leq s \Rightarrow y \in S)$ and somehow turn it into coinitial: $\forall y\in \mathbb{F}, (\exists s \in S, s \geq y)$.
I do not exactly see how one systematically creates the notion of co- for coinitial sets. And especially so for cofinal sets.
Thank you for your help.