Definition:Cut (Analysis)

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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.


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