Definition:Dedekind Cut

Definition

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

A Dedekind cut of $\struct {S, \preceq}$ 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 section in $S$)

$(2): \quad \forall x \in L: \exists y \in L: x \prec y$

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.

$(3): \quad \forall x \in L: \forall y \in R: x \prec y$.

The Dedekind cut is usually defined with respect to the set of real numbers $\R$.

A Dedekind cut is also known as a Dedekind section.

Some sources refer merely to a cut, but this is too ambiguous for $\mathsf{Pr} \infty \mathsf{fWiki}$.

Some sources refer to a separation, and subclassify into $2$ different types, but this approach adds confusion.