Dedekind cut

Work over the totally ordered field $\mathbb{Q}$ of rationals (The rationals as equivalence classes of pairs of integers, The rationals form a totally ordered field). A subset $A \subseteq \mathbb{Q}$ is a Dedekind cut iff it satisfies all three of:

• (C1) $A \ne \emptyset$ and $A \ne \mathbb{Q}$ (proper and nonempty);
• (C2) downward closed: if $p \in A$ and $q < p$, then $q \in A$;
• (C3) no greatest element: if $p \in A$, then there exists $r \in A$ with $p < r$.

This is the lower-set convention (Rudin's): a cut is the set of rationals lying strictly below a real point, so it "opens downward" and never closes off at a maximum. Under the opposite (upper-set) convention the inequalities are reversed; we fix the lower-set form throughout.

An equivalent phrasing of (C2) by contraposition: if $q \notin A$ and $q < p$, then $p \notin A$; the complement $\mathbb{Q} \setminus A$ is upward closed. Consequently every $a \in A$ and $b \notin A$ satisfy $a < b$: were $b \le a$, downward closure (C2) would place $b \in A$. Thus a cut splits $\mathbb{Q}$ into a lower piece $A$ and an upper piece $\mathbb{Q} \setminus A$ with every element of the former below every element of the latter, the lower piece having no largest member.

The set of all Dedekind cuts is the carrier of the real numbers (The real numbers $\mathbb{R}$ as Dedekind cuts); each cut is a real number, identified with the downward gap of rationals it names.