query: Prove "Any open subset of $\Bbb R$ is an at most countable union of pairwise disjoint open intervals”.
positive_passages.docid: 149317
positive_passages.text: An open subset of $\mathbb{R}$ is an at-most-countable union of disjoint open intervals
negative_passages.docid: 136006
negative_passages.text: Prove that any countable cartesian product of countable sets is countable
negative_passages.docid: 147213
negative_passages.text: Prove that finite unions of compact sets are compact
negative_passages.docid: 6491
negative_passages.text: Not every open set in $\mathbb{R}^n$, for $n\geq2$, is a disjoint union of countable open balls.
negative_passages.docid: 120273
negative_passages.text: Showing that a collection of sets is a $\sigma$-algebra: either set or complement is countable
negative_passages.docid: 95244
negative_passages.text: Prove that a finite union of closed sets is also closed
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: 68230
negative_passages.text: The cardinality of a countable union of sets with less than continuum cardinality
negative_passages.docid: 63538
negative_passages.text: Impossibility of constructing a continuum-size linearly independent set in $\Bbb R$
negative_passages.docid: 70164
negative_passages.text: Proof exercise that the cardinal number of a set in $\mathbb R^2$ is $\frak c$
negative_passages.docid: 73368
negative_passages.text: Does this proof that every open set in a metric space is a union of open balls use the axiom of choice?
negative_passages.docid: 138768
negative_passages.text: Showing that the family of Borel sets is countably generated
negative_passages.docid: 92639
negative_passages.text: Prove that the family of open sets in $\mathbb{R}$ has cardinality equal to $2^{\aleph_0}$
negative_passages.docid: 121626
negative_passages.text: Every open cover of the real numbers has a countable subcover (Lindelöf's lemma)
negative_passages.docid: 138283
negative_passages.text: $|(a,b)| = |\Bbb R|$ ? Cardinality of any open interval
negative_passages.docid: 70143
negative_passages.text: Mutually exclusive countable subsets of a countable set
negative_passages.docid: 15480
negative_passages.text: Open set in a metric space is union of closed sets