$\overline{\mathbb{R}}$ is a topological space, but not a metric space, so I'm not sure if it is true.
Let $U$ be an open set in $\overline{\mathbb{R}}$ such that $\infty\in U$.
Then, how do I that prove there exists $c\in\mathbb{R}$ such that $(c,\infty] \subset U$ and such that $(c,\infty]$ is an open set?