Dedekind cut

In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind (but previously considered by Joseph Bertrand[1][2]), are а method of construction of the real numbers from the rational numbers.

Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between A and B.

[3] In other words, A contains every rational number less than the cut, and B contains every rational number greater than or equal to the cut.

[3] Dedekind cuts can be generalized from the rational numbers to any totally ordered set by defining a Dedekind cut as 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.

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.

such that By omitting the first two requirements, we formally obtain the extended real number line.

It is more symmetrical to use the (A, B) notation for Dedekind cuts, but each of A and B does determine the other.

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".

If the ordered set S is complete, then, for every Dedekind cut (A, B) of S, the set B must have a minimal element b, hence we must have that A is the interval (−∞, b), and B the interval [b, +∞).

The important purpose of the Dedekind cut is to work with number sets that are not complete.

by putting every negative rational number in A, along with every non-negative rational number whose square is less than 2; similarly B would contain every positive rational number whose square is greater than or equal to 2.

In this way, set inclusion can be used to represent the ordering of numbers, and all other relations (greater than, less than or equal to, equal to, and so on) can be similarly created from set relations.

Moreover, the set of Dedekind cuts has the least-upper-bound property, i.e., every nonempty subset of it that has any upper bound has a least upper bound.

Thus, constructing the set of Dedekind cuts serves the purpose of embedding the original ordered set S, which might not have had the least-upper-bound property, within a (usually larger) linearly ordered set that does have this useful property.

A typical Dedekind cut of the rational numbers

, and further, by defining properly arithmetic operators over these sets (addition, subtraction, multiplication, and division), these sets (together with these arithmetic operations) form the familiar real numbers.

(please refer to the link above for the precise definition of how the multiplication of cuts is defined), is

(note that rigorously speaking this number 2 is represented by a cut

Now armed with the multiplication between cuts, it is easy to check that

Given a Dedekind cut representing the real number

This property and its relation with real numbers given only in terms of

is particularly important in weaker foundations such as constructive analysis.

In the general case of an arbitrary linearly ordered set X, a cut is a pair

[5] If neither A has a maximum, nor B has a minimum, the cut is called a gap.

The relevant notion in this case is a Cuesta-Dutari cut,[7] named after the Spanish mathematician Norberto Cuesta Dutari [es].

One completion of S is the set of its downwardly closed subsets, ordered by inclusion.

A related completion that preserves all existing sups and infs of S is obtained by the following construction: For each subset A of S, let Au denote the set of upper bounds of A, and let Al denote the set of lower bounds of A.

Then the Dedekind–MacNeille completion of S consists of all subsets A for which (Au)l = A; it is ordered by inclusion.