Dedekind CutYou are currentlynot logged in Click here to log in 


Dedekind cuts are one method of construction of the real numbers.
The set $B$ may or may not have a smallest element among the rationals. If $B$ has a smallest element among the rationals, the cut corresponds to that rational. Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between $A$ and $B.$ In other words, $A$ contains every rational number less than the cut, and $B$ contains every rational number greater than the cut. An irrational cut is equated to an irrational number which is in neither set. Every real number, rational or not, is equated to one and only one cut of rationals.
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More generally, a Dedekind cut is a partition of a totally ordered set into two nonempty parts $A$ and $B,$ such that $A$ is closed downwards (meaning that for all $a$ in $A,$ $x\le{a}$ implies that $x$ is in $A$ as well) and $B$ is closed upwards, and $A$ contains no greatest element.
It is straightforward to show that a Dedekind cut among the real numbers is uniquely defined by the corresponding cut among the rational numbers. Similarly, every cut of reals is identical to the cut produced by a specific real number (which can be identified as the smallest element of the $B$ set). In other words, the number line where every real number is defined as a Dedekind cut of rationals is a complete continuum without any further gaps.
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