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.

Proof sketch (optional): If $E$ fails to contain some point between $a<b$ in $E$, then $E$ can be separated into $E\cap(-\infty,x)$ and $E\cap(x,\infty)$. Conversely, if $E$ is an interval, any attempted separation would force a “gap,” contradicting the interval property.