Definition:Upper Closure/Set

Definition
Let $\struct {S, \preceq}$ be an ordered set or preordered set.

Let $T \subseteq S$.

The upper closure of $T$ (in $S$) is defined as:


 * $T^\succeq := \bigcup \set {t^\succeq: t \in T}$

where $t^\succeq$ denotes the upper closure of $t$ in $S$.

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

Also see

 * Definition:Upper Closure of Element


 * Definition:Lower Closure of Subset
 * Definition:Strict Upper Closure of Subset


 * Upper Closure is Closure Operator
 * Upper Closure is Smallest Containing Upper Set