Compact Set of Irrational Numbers is Nowhere Dense

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Theorem

Let $\left({\R \setminus \Q, \tau_d}\right)$ be the irrational number space under the Euclidean topology $\tau_d$.

Let $S \subseteq \R \setminus \Q$ be a compact set of $\R \setminus \Q$.


Then $S$ is nowhere dense in $\R \setminus \Q$.


Proof


Sources