query: Prob. 18, Chap. 2 in Baby Rudin: Any non-empty perfect set of real numbers which contains no rationals?
positive_passages.docid: 19954
positive_passages.text: Perfect set without rational numbers
negative_passages.docid: 23164
negative_passages.text: Why are irrational numbers uncountable and rationals countable?
negative_passages.docid: 31212
negative_passages.text: The existence of the empty set is an axiom of ZFC or not?
negative_passages.docid: 70196
negative_passages.text: Is there any well-ordered uncountable set of real numbers under the original ordering?
negative_passages.docid: 87735
negative_passages.text: Prove that the roots $\in \Bbb R$ of $x^3+x+1=0$ aren't rational without RRT
negative_passages.docid: 142008
negative_passages.text: Find all natural numbers $n$ such that ${(n+1)}^{1/2}-{(n-1)}^{1/2}$ is rational
negative_passages.docid: 70418
negative_passages.text: Theorem 3.2 Rudin -- How is 0 concluded?
negative_passages.docid: 110957
negative_passages.text: No non-trivial clopen sets in $\mathbb{R}$? How to give a direct proof?
negative_passages.docid: 4381
negative_passages.text: Proof that the set of irrational numbers is dense in reals
negative_passages.docid: 87410
negative_passages.text: Are there real numbers that are neither rational nor irrational?
negative_passages.docid: 75012
negative_passages.text: Additive group of rationals has no minimal generating set
negative_passages.docid: 86988
negative_passages.text: Cardinality of the Irrationals
negative_passages.docid: 22391
negative_passages.text: Is the set of all pairs of real numbers uncountable?
negative_passages.docid: 138977
negative_passages.text: Infinitely many rational numbers between two different real numbers
negative_passages.docid: 11405
negative_passages.text: If a nonempty set of real numbers is open and closed, is it $\mathbb{R}$? Why/Why not?
negative_passages.docid: 124448
negative_passages.text: For every real number $a$ there exists a sequence $r_n$ of rational numbers such that $r_n$ approaches $a$.
negative_passages.docid: 86914
negative_passages.text: Cardinality of the set of hyperreal numbers