# Definition:Dedekind Cut

## Definition

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

### Definition 1

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

### Definition 2

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 defined as

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

## Also known as

A Dedekind cut is also known as a Dedekind section.

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

## Also see

• Results about Dedekind cuts can be found here.

## Source of Name

This entry was named for Julius Wilhelm Richard Dedekind.