Definition:Upper Closure/Set

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

Let $T \subseteq S$.

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


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

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

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

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