Supreme property

In mathematics , the supremum property is a fundamental property of real numbers , more precisely their arrangement , and certain other ordered sets . The property says that every non-empty and upwardly bounded set of real numbers has a smallest upper bound, a supremum .

The supremum property is a form of the axiom of completeness for the real numbers and is sometimes referred to as Dedekind completeness . It can be used many basic results of real analysis to show about the intermediate value theorem , the Bolzano-Weierstrass Theorem , the extreme value theorem or the Heine-Borel Theorem . It is usually assumed as an axiom for the synthetic construction of real numbers. It is also closely related to the construction of real numbers using Dedekind's cut .

In order theory , the supremum property can be generalized to a concept of completeness for every partially ordered set . A dense , totally ordered set which fulfills the supremum property is called a linear continuum .

Formal definition

Definition of real numbers

Let be a non-empty set of real numbers. ${\displaystyle S}$

• A real number is called the upper bound for , if for all . ${\displaystyle x}$ ${\displaystyle S}$ ${\displaystyle x\geq s}$ ${\displaystyle s\in S}$
• A real number is the smallest upper bound (or the supremum ) for if an upper bound is for and for every upper bound of . ${\displaystyle x}$ ${\displaystyle S}$ ${\displaystyle x}$ ${\displaystyle S}$ ${\displaystyle x\leq y}$ ${\displaystyle y}$ ${\displaystyle S}$

The supremum property says that every non-empty set of real numbers that is bounded above must have a smallest upper bound.

Generalization to ordered sets

For each subset of a partially ordered set, one can define an upper bound and a smallest upper bound if one replaces “real number” with “element of ”. In this case it is said to have the supremum property if every upwardly bounded nonempty subset of has a smallest upper bound. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$

For example, the set of rational numbers does not fulfill the supremacy property if one assumes the usual order of the rational numbers. So did the crowd ${\displaystyle \mathbb {Q} }$

${\displaystyle \left(-{\sqrt {2}},{\sqrt {2}}\right)\cap \mathbb {Q} =\left\{x\in \mathbb {Q} :x^{2}<2\right\}\,}$

an upper bound in , but no smallest upper bound in , because the square root of two is irrational . The construction of the real numbers by means of the Dedekind cut makes use of this fact by defining the irrational numbers as the suprema of certain subsets of the rational numbers. ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} }$