give an example of an open subset whose closure is the set of real numbers

Question

Is there an example of an open subset whose closure is the set of real numbers in metric spaces?

Answer 1

$\displaystyle\bigcup_{n = 0}^{\infty}((-n-1,-n) \cup (n, n+ 1))=\mathbb{R}\backslash\mathbb{Z}$ is clearly open and $\operatorname{cl} ({\mathbb{R} \backslash \mathbb{Z}}) =\mathbb{R}$ where $\operatorname{cl} $ denotes the closure.