Least-upper-bound property

In mathematics, the Supremumseigenschaft is a fundamental property of real numbers, or more precisely their arrangement, and certain other minor quantities. The property states that every non-empty and bounded set of real numbers upward a least upper bound has a supremum.

The Supremumseigenschaft is a form of the completeness axiom for the real numbers and is sometimes called Dedekind completeness. It can be used to demonstrate many basic results of real analysis, such as the intermediate value theorem, the theorem of Bolzano - Weierstrass extreme value theorem or the theorem of Heine- Borel. For the synthetic construction of the real numbers, it is usually assumed as an axiom. With the construction of the real numbers by means of the Dedekind cut them as inextricably connected.

In order theory, the Supremumseigenschaft can be generalized to a complete concept for a lot of partially ordered. A density, totally ordered set which satisfies the Supremumseigenschaft is called linear continuum.

Formal definition

Definition of real numbers

Be a non-empty set of real numbers.

• A real number is called an upper bound for, if for all.
• A real number is the least upper bound (or supremum ) for when an upper bound for is and for every upper bound of.

The Supremumseigenschaft states that every non-empty set of real numbers which is bounded above, must have a least upper bound.

Generalization to ordered sets

One can define an upper bound and a least upper bound for each subset of a partially ordered set, if one " real number " vs. " element of" replace. In this case, they say, have the Supremumseigenschaft if every bounded above non-empty subset of a least upper bound has.

For example, satisfies the set of rational numbers, the Supremumseigenschaft not, if we assume the usual order of the rational numbers. Thus, the amount of

An upper bound in, but no least upper bound in, because the square root of two is irrational. The construction of the real numbers by means of the Dedekind section uses this fact by the irrational numbers are defined as the suprema of certain subsets of the rational numbers.

• Analysis
• Order theory