query: Proving Every open set in $\Bbb R$ is a countable union of open intervals.
positive_passages.docid: 21892
positive_passages.text: Any open subset of $\Bbb R$ is a countable union of disjoint open intervals
negative_passages.docid: 81509
negative_passages.text: The cardinality of a countable union of countable sets, without the axiom of choice
negative_passages.docid: 92526
negative_passages.text: Collection of clopen sets form a base for a countable completely regular space
negative_passages.docid: 6523
negative_passages.text: Prove that the space has a countable dense subset
negative_passages.docid: 118595
negative_passages.text: Proving that a set is countable by finding a bijection
negative_passages.docid: 59885
negative_passages.text: Proof Verification - Every sequence in $\Bbb R$ contains a monotone sub-sequence
negative_passages.docid: 70048
negative_passages.text: In which topologies do open sets maintain open under countable or arbitrary intersection?
negative_passages.docid: 55484
negative_passages.text: Proving that the map from a topological space to a fiber bundle is open.
negative_passages.docid: 12472
negative_passages.text: Let $X$ and $Y$ be countable sets. Then $X\cup Y$ is countable
negative_passages.docid: 153934
negative_passages.text: How to prove that a set R\Z is open
negative_passages.docid: 153652
negative_passages.text: Countable ordinals are embeddable in the rationals $\Bbb Q$ -- proofs and their use of AC
negative_passages.docid: 82628
negative_passages.text: Cardinality of a set of closed intervals
negative_passages.docid: 94825
negative_passages.text: Open set as a finite union of closed sets
negative_passages.docid: 35692
negative_passages.text: The set of all finite subsets of the natural numbers is countable
negative_passages.docid: 66398
negative_passages.text: Union of a finite set and a countably infinite set is countably infinite
negative_passages.docid: 82825
negative_passages.text: Proving set of interiors is open
negative_passages.docid: 43268
negative_passages.text: How to prove that the set of rational numbers are countable?