Dedekind theorem

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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 $B$.


How to Cite This Entry:
Dedekind theorem. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article