Definition:Dedekind Complete Set

This page is about Dedekind Complete Set. For other uses, see Complete.

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

• Dedekind Completeness is Self-Dual
• Definition:Dedekind Completion

• Results about Dedekind complete sets 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