Let $U \subset \mathbb{R}^n$ be nonempty and open with the [subspace topology](/page/Subspace%20Topology) inherited from $\mathbb{R}^n$. Then $U$ is connected if and only if for every $x,y \in U$, there exist an integer $m \geq 1$ and points $p_0,p_1,\dots,p_m \in U$ such that $p_0=x$, $p_m=y$, and the closed line segment $\{(1-t)p_{j-1}+tp_j : t \in [0,1]\}$ is contained in $U$ for each $j \in \{1,\dots,m\}$.