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 $\R \setminus \Q$.
Proof
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
When this page/section has been completed, {{ProofWanted}} should be removed from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
Sources
- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology ... (previous) ... (next): $\text{II}: \ 31: \ 8$
