Definition:Dedekind Completeness Property

From ProofWiki
Jump to navigation Jump to search

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


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.