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# Def/Dedekind cut

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Definition of Dedekind cut: Suppose that $(X, \leq)$ is a totally ordered set. A Dedekind cut in $X$ is an ordered pair $(L,R)$ of subsets of $X$, such that:

• Neither $L$ nor $R$ is empty.
• $L \cap R = \emptyset$.
• $L \cup R = X$.
• If $\ell \in L$, and $x \in X$, and $x \leq \ell$, then $x \in L$. ($L$ is "closed downwards").
• If $r \in R$, and $x \in X$, and $x \geq r$, then $x \in R$. ($R$ is "closed upwards").
• The set $L$ does not have a maximal element.

## Logical Connections

This definition logically relies on the following definitions and statements: Def/Total order, Def/Maximal element

The following statements and definitions logically rely on the material of this page: Def/Real number

To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Ordered ring theory

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