Let $n \in \mathbb{N}$, and equip $\mathbb{R}^n$ with its Euclidean norm $|\cdot|$ and standard topology. Let $U \subset \mathbb{R}^n$ be a nonempty open connected set. For every $x,y \in U$, there exist $m \in \mathbb{N}$ and points $p_0,\dots,p_m \in U$ such that $p_0=x$, $p_m=y$, and for every $k \in \{1,\dots,m\}$ the line segment

\begin{align*} \{(1-t)p_{k-1}+tp_k:t\in[0,1]\} \end{align*}

is contained in $U$. Equivalently, there exists a polygonal path in $U$ from $x$ to $y$.