# Dedekind completion

of a Riesz space

A Riesz space is called Dedekind complete if every non-empty subset that is bounded from below (respectively, above) has an infimum (respectively, supremum). A Dedekind-complete Riesz space is automatically Archimedean. Hence, so are its Riesz subspaces.

Given an Archimedean Riesz space $L$, a Dedekind completion of $L$ is a pair $( M,T )$ where $M$ is a Riesz space and $T : L \rightarrow M$ is a mapping such that

1) $M$ is Dedekind complete;

2) $T$ is a Riesz isomorphism of $L$ onto a Riesz subspace $T ( L )$ of $M$;

3) as a mapping $L \rightarrow M$, $T$ is normal, i.e., it preserves arbitrary suprema and infima;

4) for all $a \in M$,

$$a = \sup \left \{ {x \in T ( L ) } : {x \leq a } \right \} = \inf \left \{ {x \in T ( L ) } : {x \geq a } \right \} .$$

Every Archimedean Riesz space $L$ has a Dedekind completion, whose underlying partially ordered set can be obtained from the MacNeille completion (cf. Completion, MacNeille (of a partially ordered set)) of $L$ by removing its largest and smallest elements. The Dedekind completion is unique in the following sense. If $( M _ {1} ,T _ {1} )$ and $( M _ {2} ,T _ {2} )$ are Dedekind completions of $L$, then there exists a unique Riesz isomorphism $S$ of $M _ {1}$ onto $M _ {2}$ with $T _ {2} = S \circ T _ {1}$. More generally, if $( M,T )$ is a Dedekind completion of $L$, then every normal Riesz homomorphism of $L$ into any Dedekind-complete Riesz space $K$ can uniquely be extended to a normal Riesz homomorphism $M \rightarrow K$.

The Riesz spaces $L _ {p} ( \mu )$( $1 \leq p < \infty$) are Dedekind complete; so is $L _ \infty ( \mu )$ if $\mu$ is $\sigma$- finite. The space $C ( X )$( $X$ a compact Hausdorff space) is Dedekind complete if and only if $X$ is extremally disconnected (cf. Extremally-disconnected space). There are few non-trivial instances of Riesz spaces whose Dedekind completions are to some extent "understood" . The Dedekind completion of the space $c$ of all converging sequences is $l _ \infty$. That of $C ( X )$ is the quotient $B ( X ) /N$, where $B ( X )$ is the space of all bounded Borel functions and $N$ is the ideal of all functions that vanish off meager sets (cf. Category of a set). (In either case, the mapping $T : L \rightarrow M$ is obvious.)

How to Cite This Entry:
Dedekind completion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dedekind_completion&oldid=46597
This article was adapted from an original article by A.C.M. van Rooy (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article