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 $p \in \Q$ and $q \in \Q$ such that $q < p$, then $q \in \Q$


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

Then $\alpha$ is called a cut.