Dedekind cut

Builds on The rationals form an ordered field

Definition

A Dedekind cut is a pair $(A, B)$ of subsets of the rational numbers $\mathbb{Q}$ with these properties:

$A$ and $B$ are nonempty, $A \cap B = \emptyset$ and $A \cup B = \mathbb{Q}$ ;
if $a \in A$ , $q \in \mathbb{Q}$ and $q < a$ , then $q \in A$ ;
$A$ has no greatest element.

Remarks

The set $A$ determines the cut, because $B = \mathbb{Q} \setminus A$ . Thus some authors call $A$ itself the cut. The cut with $A = \{q \in \mathbb{Q} : q < 0 \text{ or } q^2 < 2\}$ corresponds to $\sqrt{2}$ . Here $B$ has no least element. The set of all cuts, ordered by inclusion of the lower sets, is a complete ordered field with suitable operations.

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Integers object
Ordered field definition
Rational numbers object
Ring of integers statement
The rationals form a field statement
The rationals form an ordered field statement
Dedekind cut definition

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Hölder's theorem on Archimedean groups

Tag	`04QNZZ4HHY`
URI	`m2://04QNZZ4HHY`
Kind	definition
Writing	W1 stated
MSC	26A03
Defines	Dedekind cut
Uses	The rationals form an ordered field

Dedekind cut

Definition

Remarks

Reading path

Where this leads

Further reading

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