Definition:Dedekind Cut

Definition
Let $\left({S, \preceq}\right)$ be a totally ordered set.

Let $S$ be partitioned into two subsets $L$ and $R$ such that $\forall x \in L: \forall y \in R: x \prec y$.

That is:
 * Every $s \in S$ belongs to one or the other (but not both) of the two sets $L$ and $R$
 * Each of $L$ and $R$ contains at least one element of $S$
 * Any element of $L$ strictly precedes any element of $R$.

Then the two sets $L$ and $R$ are called a section of $S$.

This concept is also known as a cut, or a Dedekind cut, for Richard Dedekind.

See Dedekind's Theorem.