Definition:Dedekind Cut/Definition 1

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

A Dedekind cut of $\left({S, \preceq}\right)$ is a non-empty proper subset $L \subsetneq S$ such that:
 * $(1): \quad \forall x \in L: \forall y \in S: y \prec x \implies y \in L$ ($L$ is a lower set in $S$)
 * $(2): \quad \forall x \in L: \exists y \in L: x \prec y$

Also see

 * Equivalence of Definitions of Dedekind Cut