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 $\R \setminus \Q$.

Proof

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Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $31$. The Irrational Numbers: $8$