Compact Set of Irrational Numbers is Nowhere Dense
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This theorem requires a proof..
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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
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Sources
- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology ... (previous) ... (next): $\text{II}: \ 31: \ 8$
