query: How can every open set in R be a countable union of pairwise disjoint open intervals in R?
positive_passages.docid: 148638
positive_passages.text: Write $(0,1)$ as a countable union of disjoint open intervals
negative_passages.docid: 59424
negative_passages.text: Showing that $B=\{(x,y) \in \Bbb{R^2} \mid \ xy>0 \}$ is an open set
negative_passages.docid: 97697
negative_passages.text: Countable union of sets of cardinality $c$ has cardinality $c$
negative_passages.docid: 68230
negative_passages.text: The cardinality of a countable union of sets with less than continuum cardinality
negative_passages.docid: 18572
negative_passages.text: Union of a family of connected pairwise not disjoint subspaces is connected
negative_passages.docid: 91736
negative_passages.text: Explicit homeomorphism between open and closed rational intervals?
negative_passages.docid: 18223
negative_passages.text: Prove that the union of two disjoint countable sets is countable
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: 103769
negative_passages.text: The Cartesian product of a finite number of countable sets is countable
negative_passages.docid: 149317
negative_passages.text: An open subset of $\mathbb{R}$ is an at-most-countable union of disjoint open intervals
negative_passages.docid: 155311
negative_passages.text: How to get a set $E$ such that $0<\mu(E\cap I)<\mu(E)$ for all interval $I$?
negative_passages.docid: 82628
negative_passages.text: Cardinality of a set of closed intervals
negative_passages.docid: 101979
negative_passages.text: Set of open intervals in R with rational endpoints is a basis for standard topology on R
negative_passages.docid: 144731
negative_passages.text: Disjoint closed sets in a second countable zero-dimensional space can be separated by a clopen set
negative_passages.docid: 88079
negative_passages.text: R has an uncountable number of distinct dense subsets.
negative_passages.docid: 23442
negative_passages.text: Cardinality of the union of infinite and countable sets
negative_passages.docid: 62317
negative_passages.text: Generating set of countable, co-countable sigma algebra on $\mathbb{R}$