Definition:Dedekind Cut

Definition 1
Let $\left({S, \preceq}\right)$ be a totally ordered set.

A Dedekind cut of $S$ 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.
 * $(3): \quad \forall x \in L: \forall y \in R: x \prec y$.

Definition 2
Let $\left({S, \preceq}\right)$ be a totally ordered set.

A Dedekind cut of $S$ is a non-empty proper subset $L \subsetneq S$ such that:
 * $(1): \quad L$ does not have a greatest element.
 * $(2): \quad \forall x \in L: \mathop \downarrow \left({x}\right) \subseteq L$, where $\mathop \downarrow \left({x}\right)$ denotes the strict lower closure of $x$ in $S$.

Also known as
A Dedekind cut is also known as a Dedekind section.

Also see

 * Dedekind's Theorem
 * Dedekind Complete