Theorems · Theorem · general topology
TotallyBounded.isCompact_of_isClosed
∀ {α : Type u} [uniformSpace : UniformSpace α] [CompleteSpace α] {s : Set α},
TotallyBounded s → IsClosed s → IsCompact s- Defined in
- Mathlib.Topology.UniformSpace.Cauchy
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- UniformSpaceCompleteSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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
- CompleteSpacestatement and proof · cited by 2,532
- UniformSpacestatement and proof · cited by 2,040
- IsClosedstatement and proof · cited by 1,639
- IsCompactstatement · cited by 1,282
- TotallyBoundedstatement and proof · cited by 79
- IsClosed.isCompleteproof · cited by 19
- TotallyBounded.isCompact_of_isCompleteproof · cited by 2
Cited by5
Results whose statement or proof uses this declaration.
- isClosed_setOfPred_isCompactOperatorproof · cited by 3
- Filter.TotallyBounded.isCompact_setOfPred_clusterPtproof · cited by 1
- TopologicalSpace.Compacts.isClosedEmbedding_toClosedsproof · cited by 1
- BoundedContinuousFunction.arzela_ascoli₁proof · cited by 1
- MeasureTheory.isTightMeasureSet_of_isCompact_closureproof · cited by 0