Definition:Dedekind Completion

Definition
Let $S$ be an ordered set.

A Dedekind completion of $S$ is a Dedekind complete ordered set $\tilde{S}$ together with an increasing mapping $\phi: S \to \tilde{S}$, subject to:


 * For all Dedekind complete ordered sets $X$, and for all increasing mappings $f: S \to X$, there exists a unique increasing mapping $\tilde{f}: \tilde{S} \to X$ such that:


 * $\tilde{f} \circ \phi = f$

Also see

 * Existence of Dedekind Completion
 * Dedekind Completion is Unique up to Unique Isomorphism

This concept is not to be confused with the Dedekind–MacNeille completion.