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