Definition:Dedekind Cut

Definition 1
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:
 * $({1}): \quad \forall x \in L: \forall y \in S: y \prec x \implies y \in L$
 * $({2}): \quad \forall x \in L: \exists y \in L: x \prec y$

Definition 2
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 known as
A Dedekind cut is also known as a Dedekind section.

Also see

 * Dedekind's Theorem
 * Dedekind Complete