Dedekind theorem

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