\begin{definition}[Definition:Cut (Analysis)]
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'''.
\end{definition}