# Compact Set of Rational Numbers is Nowhere Dense

## Theorem

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

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

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

## Proof

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 30: \ 8$