Intervals

Definition: Interval

Let $S$ be a set with a Total Order $\le$ and a Strict Total Order $<$.

Every pair $a,b\in S$ with $a<b$ defines:

• an open interval $(a;b)$ as the set $\{x\in S\mid a<x<b\}$;
• a half-open interval $(a;b]$ as the set $\{x\in S\mid a<x\le b\}$;
• a half-open interval $[a;b)$ as the set $\{x\in S\mid a\le x<b\}$
• a closed interval $[a;b]$ as the set $\{x\in S\mid a\le x\le b\}$.