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Dedekind theorem

From Encyclopedia of Mathematics
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on the continuity of the real axis

For any cut $A|B$ of the set of real numbers (see Dedekind cut) there exists a real number $\alpha$ which is either the largest in the class $A$ or the smallest in the class $B$. This statement is also known as the Dedekind principle (axiom) of continuity of the real axis (cf. Real number). 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: http://encyclopediaofmath.org/index.php?title=Dedekind_theorem&oldid=32026
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article