Dedekind cut

Definition

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

1. $A$ and $B$ are nonempty, $A \cap B = \emptyset$ and $A \cup B = \mathbb{Q}$;
2. if $a \in A$, $q \in \mathbb{Q}$ and $q < a$, then $q \in A$;
3. $A$ 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.

Explanations

L1 human

Cut the rational number line at one point. Put the numbers to the left in $A$ and the numbers to the right in $B$. When the cut is at a gap of $\mathbb{Q}$, such as $\sqrt{2}$, the cut itself stands for the missing number.