Definition:Upper Closure/Set

Definition
Let $\left({S, \preceq}\right)$ be an ordered set or preordered set.

Let $T \subseteq S$.

Then we define:


 * ${\uparrow} T := \bigcup \left\{{ {\bar\uparrow} t: t \in T }\right\}$

where ${\bar\uparrow} t$ is the upper closure of $t$.

That is:
 * ${\uparrow} T := \{ u \in S: \exists t \in T: t \preceq u \}$

${\uparrow} T$ is described as the upper closure of $T$ (in $S$).

Remark
The notation ${\uparrow} a$ may also be used to refer to the upper closure of an element of an ordered set.

This usage, however, clashes with a notation currently common on for the strict up-set of an element, so the upper closure of an element $a$ should be written ${\bar\uparrow} a$.

Also see

 * Definition:Lower Closure/Set
 * Definition:Strict Up-Set/Set
 * Upper Closure is Closure Operator
 * Upper Closure is Smallest Containing Upper Set