Order completeness

Order completeness is a term from algebra , especially from body theory , but which can be defined for any ordered set . The concept of order completeness turns out to be related to the concept of completeness in metric spaces in order topology for not too “large” ordered sets .

definition

An order on is called order-complete if one of the following equivalent conditions applies: ${\displaystyle X}$

• Every non-empty, downwardly bounded subset has an infimum .
• Every non-empty upwardly restricted subset has a supremum . (The so-called supremum property .)
• Every non-empty limited set has an infimum and a supremum.

Relation to metric completeness

If the order topology is metrizable , then the order is order- complete if and only if it is completely metrizable , i. H. if there is a metric that creates the order topology and makes it a complete metric space . ${\displaystyle X}$ ${\displaystyle (X,<)}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$

Neat body

The concept of order completeness is particularly important in the theory of ordered bodies . It enables the following characterization of the field of real numbers :

An ordered body is isomorphic to if and only if it is orderly complete. ${\displaystyle \mathbb {R} }$

literature

• Herbert Amann, Joachim Escher: Analysis I . Springer, 3rd edition, 2006, ISBN 9783764377564 , p. 98
• Hermann Schichl, Roland Steinbauer: Introduction to mathematical work . Springer, 2012, ISBN 9783642286452 , pp. 316-320
• AH Lightstone: Linear Algebra . Appleton-Century-Crofts, 1969 pp. 178-180

Web links

• R: Verfürth: Analysis I . Script, Ruhr-Universität Bochum, pp. 35–42, especially 42
• The Complete Ordered Field: The Real Numbers
• John J. O'Connor: Axioms for the Real numbers - Chapter of an analysis sprite from the University of St Andrews

Individual evidence

1. ↑ K.-U. Bux: Analysis I , Theorem 8.4
2. ↑ D. Lenz: Analysis I , Chapter 2.4