Definition:Supremum of Set

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

Let $T \subseteq S$.

An element $c \in S$ is the supremum of $T$ in $S$ :


 * $(1): \quad c$ is an upper bound of $T$ in $S$
 * $(2): \quad c \preceq d$ for all upper bounds $d$ of $T$ in $S$.

The supremum of $T$ is denoted $\sup T$.

If there exists a supremum of $T$ (in $S$), we say that:
 * $T$ admits a supremum (in $S$) or
 * $T$ has a supremum (in $S$).

Subset of Real Numbers
The concept is usually encountered where $\left({S, \preceq}\right)$ is the set of real numbers under the usual ordering: $\left({\R, \le}\right)$:

Also see

 * Definition:Infimum of Set


 * Supremum and Infimum are Unique