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

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

• $\mathbb R$ itself? $\mathbb R-\{0\}$? – lulu Mar 23 at 18:17
• A less trivial example is $\{x\in\mathbb R\;|\; x-\lfloor x\rfloor\notin C\}$ where $C$ is the Cantor set on $[0,1]$. – Don Thousand Mar 23 at 18:38

## 1 Answer

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