Compact Set of Irrational Numbers is Nowhere Dense

Theorem

Let $\struct {\R \setminus \Q, \tau_d}$ 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 $\R \setminus \Q$.