Definition:Cut (Analysis)

Definition

Let $\alpha \subset \Q$ be a subset of the set of rational numbers $\Q$ which has the following properties:

$(1): \quad \alpha \ne \O$ and $\alpha \ne \Q$, that is: $\alpha$ contains at least one rational number but not all rational numbers

$(2): \quad$ If $p \in \alpha$ and $q \in \Q$ such that $q < p$, then $q \in \alpha$

$(3): \quad \alpha$ does not contain a greatest element.

Then $\alpha$ is called a cut.

Lower Number

Let $\alpha$ be a cut.

Let $p \in \alpha$.

Then $p$ is referred to as a lower number of $\alpha$.

Upper Number

Let $\alpha$ be a cut.

Let $q \in \Q$ such that $q \notin \alpha$.

Then $p$ is referred to as an upper number of $\alpha$.

Rational Cut

Let $r \in \Q$ be a rational number.

Let $\alpha$ be the cut consisting of all rational numbers $p$ such that $p < r$.

Then $\alpha$ is referred to as a rational cut.

Also see

• Results about cuts can be found here.

Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Dedekind Cuts: $1.4$. Definition