Definition:Dedekind Cut/Definition 1

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 $S$)

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

Also see

• Equivalence of Definitions of Dedekind Cut

Source of Name

This entry was named for Julius Wilhelm Richard Dedekind.