Dedekind cut
Noun
Meaning
Dedekind cut (plural Dedekind cuts)
(mathematics) Any partition of the set of rational numbers into non-empty sets A and B such that all elements of A are less than all elements of B and A contains no greatest element; intended as a construction of a real number.
Examples
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.
It can be a simplification, in terms of notation if nothing more, to concentrate on one "half" — say, the lower one — and call any downward closed set A without greatest element a "Dedekind cut".
It is straightforward to show that a Dedekind cut among the real numbers is uniquely defined by the corresponding cut among the rational numbers.
The important purpose of the Dedekind cut is to work with number sets that are not complete.