Definition:Dedekind Complete Set

Definition

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

Then $\left({S, \preceq}\right)$ is Dedekind complete if and only if every non-empty subset of $S$ that is bounded above admits a supremum (in $S$).

Also known as

This is commonly referred to as:

the supremum property
the least upper bound property
the infimum property
the greatest lower bound property

where the latter denominations are justified by Dedekind Completeness is Self-Dual.

Some sources hyphenate: Dedekind-complete. In the interest of consistency, $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers the non-hyphenated version.

Also see

• Results about Dedekind complete sets can be found here.

Source of Name

This entry was named for Julius Wilhelm Richard Dedekind.