Theorems · Theorem · general topology
TopologicalSpace.IsSeparable.separableSpace
∀ {X : Type u_2} [inst : TopologicalSpace X] [TopologicalSpace.PseudoMetrizableSpace X] {s : Set X},
TopologicalSpace.IsSeparable s → TopologicalSpace.SeparableSpace ↑sIf a set s is separable, then the corresponding subtype is separable in a
pseudo metrizable space.
This is not obvious, as the countable set whose closure covers s does not need in
general to be contained in s.
- Defined in
- Mathlib.Topology.Metrizable.Basic
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Set.Elemstatement and proof · cited by 7,166
- Set.imageproof · cited by 5,609
- closureproof · cited by 1,254
- Set.Countableproof · cited by 545
- Function.Injective.injOnproof · cited by 280
- TopologicalSpace.PseudoMetrizableSpacestatement and proof · cited by 245
- Subtype.coe_injectiveproof · cited by 205
- TopologicalSpace.SeparableSpacestatement · cited by 109
- TopologicalSpace.IsSeparablestatement and proof · cited by 51
- Topology.IsInducing.subtypeValproof · cited by 43
Cited by5
Results whose statement or proof uses this declaration.
- MeasureTheory.StronglyMeasurable.separableSpace_range_union_singletonproof · cited by 11
- MeasureTheory.Lp.simpleFunc.denseRange_coeSimpleFuncNonnegToLpNonnegproof · cited by 1
- ContinuousOn.isSeparable_imageproof · cited by 1
- Topology.IsInducing.isSeparable_preimageproof · cited by 1
- Convex.integral_memproof · cited by 1