Let $U$ be a non-empty open subset of $\mathbb{R}$. Suppose that there exists a uniformly continuous homeomorphism $h : U \longrightarrow \mathbb{R}$. Show that $U = \mathbb{R}$.
Let $U$ be a non-empty open subset of $\mathbb{R}$. Suppose that there exists a uniformly continuous homeomorphism $h : U \longrightarrow \mathbb{R}$. Show that $U = \mathbb{R}$.