Connectedness criteria in R

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

$E$ is connected.
$E$ is an interval, with the empty set and one-point sets allowed.
$E$ is order-convex: whenever $a,b\in E$ and $a<c<b$ , one has $c\in E$ .

Applications

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.