# Dedekind theorem

From Encyclopedia of Mathematics

*on the continuity of the real axis*

For any cut of the set of real numbers (see Dedekind cut) there exists a real number which is either the largest in the class or the smallest in the class . This statement is also known as the Dedekind principle (axiom) of continuity of the real axis (cf. Real number). The number is the least upper bound of and the greatest lower bound of .

**How to Cite This Entry:**

Dedekind theorem.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Dedekind_theorem&oldid=17055

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