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.