Connectedness criteria in R

Connectedness criteria in $\mathbb{R}$: Let $E\subseteq \mathbb{R}$ with the usual topology. The following are equivalent:

1. $E$ is connected .
2. $E$ is an interval .
3. (Intermediate-point property) If $a,b\in E$ with $a<b$ and $c\in\mathbb{R}$ satisfies $a<c<b$, then $c\in E$.
4. (No gap) There do not exist real numbers $a<b$ such that $E\cap(-\infty,a)$ and $E\cap(b,\infty)$ are both nonempty while $E\cap(a,b)=\varnothing$.

These criteria are all manifestations of the same phenomenon: in $\mathbb{R}$, connectedness is completely controlled by order. They are often paired with continuity preserves connectedness to pin down real-valued images.