Category:Definitions/Dedekind Cuts

This category contains definitions related to Dedekind Cuts.
Related results can be found in Category:Dedekind Cuts.

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.

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