# Compact Set of Irrational Numbers is Nowhere Dense

## 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 $\R \setminus \Q$.