Theorems · Theorem · general topology
TopologicalSpace.exists_dense_seq
∀ (α : Type u) [t : TopologicalSpace α] [TopologicalSpace.SeparableSpace α] [Nonempty α], ∃ u, DenseRange u
A nonempty separable space admits a sequence with dense range. Instead of running cases on the
conclusion of this lemma, you might want to use TopologicalSpace.denseSeq and
TopologicalSpace.denseRange_denseSeq.
If α might be empty, then TopologicalSpace.exists_countable_dense is the main way to use
separability of α.
- Defined in
- Mathlib.Topology.Bases
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Set.rangeproof · cited by 4,705
- Set.Countableproof · cited by 545
- Denseproof · cited by 359
- DenseRangestatement · cited by 164
- TopologicalSpace.SeparableSpacestatement and proof · cited by 109
- Dense.monoproof · cited by 43
- TopologicalSpace.exists_countable_denseproof · cited by 15
- Set.countable_iff_exists_subset_rangeproof · cited by 3
Cited by6
Results whose statement or proof uses this declaration.
- TopologicalSpace.denseSeqproof · cited by 11
- TopologicalSpace.denseRange_denseSeqproof · cited by 5
- exists_nat_nat_continuous_surjective_of_completeSpaceproof · cited by 1
- exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAtproof · cited by 1
- MeasureTheory.SeparableSpace.exists_measurable_partition_diam_leproof · cited by 1
- MeasureTheory.isTightMeasureSet_of_isCompact_closureproof · cited by 0