Definition:Dedekind Cut/Definition 2

Definition
Let $\struct {S, \preceq}$ be a totally ordered set.

A Dedekind cut of $\struct {S, \preceq}$ is an ordered pair $\tuple {L, R}$ such that:
 * $(1): \quad \set {L, R}$ 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