# Definition:Dedekind Complete Set

## Definition

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

Then $\struct {S, \preceq}$ 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.
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