Complete Ordering

Last updated Nov 6, 2022

Let $(T, \leq)$ be a Total Ordering on $T$. Then $\leq$ is a Complete Ordering relation if for every Nonempty Set $A \subset T$ s.t. there exists $x \in T$ for which $A \leq x$ ($y \leq x$ $\forall y \in A$), then $\sup A$ exists in $T$.

Also known as the Least Upper Bound Property.

• Supremum
• Upper Bound