Interval

An interval is a subset $I\subseteq\mathbb R$ such that whenever $a,c\in I$ and $a<b<c$, then $b\in I$.

Intervals are the basic “connected pieces” of the ordered real line (using the order axioms ). In the usual topology on $\mathbb R$ coming from the metric $d(x,y)=|x-y|$, open intervals are fundamental examples of open sets .

Examples:

• $(0,1)=\{x\in\mathbb R: 0<x<1\}$ is an interval.
• $[0,\infty)=\{x\in\mathbb R: x\ge 0\}$ is an interval.