# The Term "Cofinal"

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.

• @Arthur: They're not the same in general. Consider the poset $\mathbb{R} \cup \{ \star \}$, where $\mathbb{R}$ has its usual order and $\star$ is not comparable to any $x \in \mathbb{R}$ (so $\star \le \star$ but, for all $x \in \mathbb{R}$, $\star \not\le x$ and $x \not\le \star$). In this poset, $\mathbb{R}$ is unbounded but not cofinal, since any cofinal subset must contain $\star$. Oct 1, 2017 at 19:12
• "co-" here means the same thing as in "co-founder" or "co-conspirator" or "co-author" or "co-owner". $\qquad$ Mar 4, 2018 at 0:28