Theorems · Theorem · general topology
completeSpace_iff_isComplete_range
∀ {α : Type u} {β : Type v} [inst : UniformSpace α] [inst_1 : UniformSpace β] {f : α → β},
IsUniformInducing f → (CompleteSpace α ↔ IsComplete (Set.range f))- Cited by
- 11 results in Mathlib
- Foundations
- Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- UniformSpaceUniformSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- Set.rangestatement and proof · cited by 4,705
- CompleteSpacestatement · cited by 2,532
- UniformSpacestatement and proof · cited by 2,040
- Set.image_univproof · cited by 322
- IsUniformInducingstatement and proof · cited by 128
- IsCompletestatement and proof · cited by 68
- isComplete_image_iffproof · cited by 5
- completeSpace_iff_isComplete_univproof · cited by 5
Cited by11
Results whose statement or proof uses this declaration.
- completeSpace_coe_iff_isCompleteproof · cited by 6
- IsUniformInducing.completeSpace_congrproof · cited by 6
- IsUniformInducing.isComplete_rangeproof · cited by 4
- Topology.IsClosedEmbedding.polishSpaceproof · cited by 2
- IsUniformInducing.completeSpaceproof · cited by 1
- Topology.IsClosedEmbedding.IsCompletelyPseudoMetrizableSpaceproof · cited by 1
- ContinuousMultilinearMap.completeSpaceproof · cited by 1
- Topology.IsClosedEmbedding.IsCompletelyMetrizableSpaceproof · cited by 1
- CompleteSpace.iInfproof · cited by 1
- UniformConvergenceCLM.completeSpaceproof · cited by 1
- ContinuousAlternatingMap.completeSpaceproof · cited by 0