Let $n \in \mathbb N$, and let $U \subset \mathbb R^n$ be open. Equip $U$ with the [subspace topology](/page/Subspace%20Topology) inherited from $\mathbb R^n$. Then $U$ is locally path-connected: for every $x \in U$ and every open subset $V \subset U$ with $x \in V$, there exists an open path-connected subset $W \subset U$ such that $x \in W \subset V$.