Let $n\in\mathbb{N}$, and let $U \subset \mathbb{R}^n$ be a nonempty open connected subset with respect to the Euclidean topology. Then $U$ is path connected: for every $x,y \in U$, there exists a continuous map $\gamma:[0,1]\to U$, where $[0,1]$ has the [subspace topology](/page/Subspace%20Topology) inherited from $\mathbb{R}$ and $U$ has the subspace topology inherited from $\mathbb{R}^n$, such that $\gamma(0)=x$ and $\gamma(1)=y$.