# Dedekind theorem

From Encyclopedia of Mathematics

*on the continuity of the real axis; Dedekind principle, Dedekind axiom*

A form of the continuity axiom for the real number system in terms of Dedekind cuts. It states that for any cut $A|B$ of the set of real numbers there exists a real number $\alpha$ which is either the largest in the class $A$ or the smallest in the class $B$. The number $\alpha$ is the least upper bound of $A$ and the greatest lower bound of $B$.

#### References

- Richard Dedekind, "Essays on the Theory of Numbers" (tr. W.W.Beman) Dover (1963) [1901]
**ISBN**0-486-21010-3 Zbl 32.0185.01 Zbl 0112.28101

**How to Cite This Entry:**

Dedekind theorem.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Dedekind_theorem&oldid=54634

This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article