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

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

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


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