Definition:Dedekind Complete Set

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

Then $\struct {S, \preceq}$ is Dedekind complete 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, prefers the non-hyphenated version.

Also see

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