Connected subsets of R are intervals

Connected subsets of $\mathbb{R}$ are intervals: A set $E\subseteq \mathbb{R}$ is connected (in the usual metric ) if and only if it is an interval in the order sense: whenever $a,b\in E$ with $a<b$, then $(a,b)\subseteq E,$ equivalently, every $x$ with $a\le x\le b$ lies in $E$.

This characterizes connectedness on the line and is the key to the intermediate value theorem and many “no gaps” arguments.