Dedekind cut

A Dedekind cut in an ordered field is a partition of it, (A, B), such that A is closed downwards (meaning that whenever a is in A and x ≤ a, then x is in A as well), B is closed upwards and A has no maximum.

The Dedekind cut is named after Richard Dedekind, who invented this construction in order to represent the real numbers as Dedekind cuts of the rational numbers. A typical Dedekind cut of the rational numbers is given by A = { a in Q : a² < 2 }, B = { b in Q : b² ≥ 2 }. This cut represents the real number √ 2 in Dedekind's construction.

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See also:

• Cauchy sequence