View source for Dedekind cut

{{TEX|done}}
''cut''

A subdivision of the set of real (or only of the rational) numbers (of) $\mathbf R$ into two non-empty sets $A$ and $B$ whose union is $\mathbf R$, such that $a<b$ for every $a\in A$ and $b\in B$. A Dedekind cut is denoted by the symbol $A|B$. The set $A$ is called the lower class, while the set $B$ is called the upper class of $A|B$. Dedekind cuts of the set of rational numbers are used in the construction of the theory of real numbers (cf. [[Real number|Real number]]). The concept of continuity of the real axis can be formulated in terms of Dedekind cuts of real numbers.

====Comments====
For the construction of $\mathbf R$ from $\mathbf Q$ using cuts see [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,   "Principles of mathematical analysis" , McGraw-Hill  (1953)</TD></TR></table>

====Comments====
More generally we may define a cut in any [[totally ordered set]] $X$ to be a partition of $X$ into two non-empty sets $A$ and $B$ whose union is $X$, such that $a<b$ for every $a\in A$ and $b\in B$: a Dedekind cut is a cut in which either $A$ has a maximal element or $B$ has a minimal element.  A [[continuous set]] is a totally ordered set in which all cuts are Dedekind cuts.

See also: [[Completion, MacNeille (of a partially ordered set)|Dedekind–MacNeille completion]].

[[Category:Order, lattices, ordered algebraic structures]]