Definition:Dedekind Cut/Definition 2

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

A Dedekind cut of $\left({S, \preceq}\right)$ is an ordered pair $\left({L, R}\right)$ such that:
 * $(1): \quad \left\{{L, R}\right\}$ is a partition of $S$.
 * $(2): \quad L$ does not have a greatest element.
 * $(3): \quad \forall x \in L: \forall y \in R: x \prec y$.

Also see

 * Equivalence of Definitions of Dedekind Cut