# Definition:Dedekind Completeness Property

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*This page is about Dedekind Completeness Property. For other uses, see Complete.*

## Definition

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

Then $\struct {S, \preceq}$ has the **Dedekind completeness property** if and only if every non-empty subset of $S$ that is bounded above admits a supremum (in $S$).

## Also known as

The **Dedekind completeness property** is commonly referred to as:

- the
**supremum property** - the
**least upper bound property** - the
**infimum property** - the
**greatest lower bound property** - the
**completeness property**

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

A set which fulfils the **Dedekind completeness property** is described as being **Dedekind complete**.

Some sources hyphenate: **Dedekind-complete**.

In the interest of consistency, $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers the non-hyphenated version.

## Also see

- Results about
**the Dedekind completeness property**can be found**here**.

## Source of Name

This entry was named for Julius Wilhelm Richard Dedekind.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**Dedekind-complete** - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations