Connected subsets of R are intervals

Connected subsets of $\mathbb{R}$ are intervals: View $\mathbb{R}$ with its usual topology. If $E\subseteq \mathbb{R}$ is connected, then $E$ is an interval: whenever $a,b\in E$ with $a<b$ and $c\in\mathbb{R}$ satisfies $a<c<b$ , one has $c\in E$ . Conversely, every interval in $\mathbb{R}$ is connected.

Application

This classification is frequently combined with continuous images preserve connectedness to identify images of connected sets under real-valued continuous functions.

Examples

The usual topology on $\mathbb R$ is the topology induced by the standard metric $d(x,y)=|x-y|$ .