Page "Dedekind cut" Paragraph 2

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.