Definition:Empty Supremum

Definition

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

Then the empty supremum is the supremum $\sup \O$.

Notation

The empty supremum is sometimes denoted as $0$.

Also see

• Supremum of Empty Set is Smallest Element: the empty supremum exists if and only if $\struct {S, \preceq}$ has a smallest element.

• Definition:Supremum of Set
• Definition:Empty Infimum

Sources

• 1982: Peter T. Johnstone: Stone Spaces ... (previous) ... (next): Chapter $\text I$: Preliminaries, Definition $1.2$