Compact element

Remarks

In a complete lattice $L$, an element $c$ is compact if and only if for every $S \subseteq L$ with $c \le \bigvee S$ there is a finite $F \subseteq S$ with $c \le \bigvee F$. Compact elements are also called finite elements, and in domain theory $c$ is compact exactly when $c \ll c$ for the way-below relation. Example: in the power set of a set $X$, a finite set $c$ is compact. If $c \subseteq \bigcup_{d \in D} d$ for a directed family $D$, each of the finitely many points of $c$ lies in some member of $D$. Directedness then gives one member containing all of them. Non-example: in the power set of $\N$, the set $\N$ is not compact. It is the union of the directed family of sets $\set{0, \ldots, n}$ and lies in none of them. The least element, when it exists, is always compact.