Dedekind Cut

In mathematics, a Dedekind cut, named after Richard Dedekind, is a partition of the rational numbers into two non-empty parts A and B, such that all elements of A are less than all elements of B, and A contains no greatest element.

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.

Whenever, then, we have to do with a cut produced by no rational number, we create a new irrational number, which we regard as completely defined by this cut ... . From now on, therefore, to every definite cut there corresponds a definite rational or irrational number .... —Richard Dedekind, Continuity and Irrational Numbers, Section IV

More generally, a Dedekind cut is a partition of a totally ordered set into two non-empty parts (A and B), such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards, and A contains no greatest element. See also completeness (order theory).

In particular, as discussed below, Dedekind cuts among the real numbers may be considered defined as cuts among the rationals. It turns out that 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.

Dedekind used the German word Schnitt (cut) in a visual sense rooted in Euclidean geometry. When two straight lines cross, one is said to cut the other. Dedekind's construction of the number line ensures that two crossing lines always have one point in common because each of them defines a Dedekind cut on the other.