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I'm trying to understand this using the definition that the interior of the closure must be non-empty for a nowhere dense set. Thanks

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Every singleton subset of $\Bbb R$ is nowhere dense in $\Bbb R$. Let $x\in\Bbb R$ be arbitrary. Then $\{x\}$ is a closed set, so $\operatorname{cl}\{x\}=\{x\}$. Clearly $\{x\}$ does not contain any non-empty open interval, so $\operatorname{int}\{x\}=\varnothing$, and $\{x\}$ is therefore nowhere dense in $\Bbb R$.

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