Difference between revisions of "Dedekind theorem"
From Encyclopedia of Mathematics
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''on the continuity of the real axis'' | ''on the continuity of the real axis'' | ||
| − | For any cut | + | For any cut $A|B$ of the set of real numbers (see [[Dedekind cut|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|Real number]]). The number $\alpha$ is the least upper bound of $A$ and the greatest lower bound of $B$. |
Revision as of 14:18, 1 May 2014
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
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