Definition:
- Consider a partially ordered set $(\mathcal X,\leq)$ with a top element $\top$.
- Call an element proper when it is not equal to $\top$. If $X'\leq X$ then say that $X$ is above $X'$.
- Call a subset $\mathcal X'\subseteq\mathcal X$ up-directed when every $X,X'\in\mathcal X'$ has an upper bound in $\mathcal X'$.
Now for something less familiar:
Definition: Call $C\in\mathcal X$ cotopen when $\{X\in\mathcal X \mid C\leq X<\top\}$ is up-directed.
In words: $C$ is cotopen when the set of proper elements above $C$ is directed.
In the special case that $\mathcal X$ is finite, it is easy to see that $C$ being cotopen is equivalent to $C$ being beneath a (unique) greatest proper element. That is, there exists some unique maximal $C\leq M<\top$.
Question: Has cotopenness been defined and studied, either for its own sake in order theory or arising as a useful property in ordered structures, and if so can anyone provide references?
Thank you.