无理数集不能表示成$\Bbb R$中的可数闭集的并

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A Problem Book in Real Analysis - Aksoy, Khamsi - Page 204
Problem 10.49 Use Baire’s Category Theorem to show that the set of all rationals $\Bbb Q$ is not the intersection of a countable collection of open sets. Use this result to show that the set of irrationals is not the union of a countable collection of closet sets.

Czhang271828

BCT: Any countable union of closed sets with empty interior has an empty interior.

Hint: Clearly, Q is coutable union of one-point sets (closed sets), thus the set of irrationals is not the union of a countable collection of closet sets.