query: Partitioning the Reals into two Locally Uncountable, Dense Sets
positive_passages.docid: 158018
positive_passages.text: Everywhere Super Dense Subset of $\mathbb{R}$
negative_passages.docid: 17461
negative_passages.text: Show that a set is dense in $[-1,1]$
negative_passages.docid: 92526
negative_passages.text: Collection of clopen sets form a base for a countable completely regular space
negative_passages.docid: 79974
negative_passages.text: The cardinality of Lebesgue sets
negative_passages.docid: 144362
negative_passages.text: The construction of a Vitali set
negative_passages.docid: 70143
negative_passages.text: Mutually exclusive countable subsets of a countable set
negative_passages.docid: 3217
negative_passages.text: Construction of uncountably many non-isomorphic linear (total) orderings of natural numbers
negative_passages.docid: 153651
negative_passages.text: Countable partition in atoms
negative_passages.docid: 30105
negative_passages.text: Multiples of an irrational number forming a dense subset
negative_passages.docid: 90826
negative_passages.text: Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly
negative_passages.docid: 119649
negative_passages.text: Every metrizable space with a countable dense subset has a countable basis
negative_passages.docid: 121831
negative_passages.text: Proving that rational numbers are dense
negative_passages.docid: 102250
negative_passages.text: Subset of $C[0,1]$ is nowhere dense
negative_passages.docid: 110957
negative_passages.text: No non-trivial clopen sets in $\mathbb{R}$? How to give a direct proof?
negative_passages.docid: 9268
negative_passages.text: Proving that the Union of Two Compact Sets is Compact
negative_passages.docid: 88595
negative_passages.text: Prob. 21, Sec. 17 in Munkres' TOPOLOGY, 2nd ed: Closure and complementation can result in at most 14 different sets
negative_passages.docid: 66334
negative_passages.text: How is every subset of the set of reals with the finite complement topology compact?