query: proofs of floor topology, dense subsets and countable bases.
positive_passages.docid: 123208
positive_passages.text: Show that the Sorgenfrey line does not have a countable basis.
negative_passages.docid: 6210
negative_passages.text: Countable compact spaces as ordinals
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: 59289
negative_passages.text: Nested Sequence of Topological Space and the Coherent Topology; Munkres Topology 35.9
negative_passages.docid: 121831
negative_passages.text: Proving that rational numbers are dense
negative_passages.docid: 119312
negative_passages.text: Product and Box Topologies
negative_passages.docid: 108226
negative_passages.text: Are these bases for a topology?
negative_passages.docid: 150796
negative_passages.text: Subspace topology and order topology (2)
negative_passages.docid: 97064
negative_passages.text: Dense set in an infinite cofinite space
negative_passages.docid: 81509
negative_passages.text: The cardinality of a countable union of countable sets, without the axiom of choice
negative_passages.docid: 135025
negative_passages.text: Axiomatization of a Topology using the Boundary
negative_passages.docid: 99478
negative_passages.text: countably infinite union of countably infinite sets is countable
negative_passages.docid: 83842
negative_passages.text: Alternate proofs (other than diagonalization and topological nested sets) for uncountability of the reals?
negative_passages.docid: 73154
negative_passages.text: Compact subset in colimit of spaces
negative_passages.docid: 120685
negative_passages.text: Topology induced from Kuratowski Closure properties
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: 52429
negative_passages.text: Explicit countable elementary extension of $\mathbb{N}$