Theorems · Theorem · general topology
IsCompact.uniformContinuousOn_of_continuous
∀ {α : Type u_1} {β : Type u_2} [inst : UniformSpace α] [inst_1 : UniformSpace β] {s : Set α} {f : α → β},
IsCompact s → ContinuousOn f s → UniformContinuousOn f sHeine-Cantor: a continuous function on a compact set of a uniform space is uniformly continuous.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 85 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.
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
- UniformSpacestatement and proof · cited by 2,040
- ContinuousOnstatement and proof · cited by 1,411
- IsCompactstatement and proof · cited by 1,282
- continuousOn_iff_continuous_domRestrictproof · cited by 51
- UniformContinuousOnstatement · cited by 47
- uniformContinuousOn_iff_restrictproof · cited by 8
- CompactSpace.uniformContinuous_of_continuousproof · cited by 5
Cited by3
Results whose statement or proof uses this declaration.
- ContinuousMap.dense_setOfPred_contDiffproof · cited by 1
- ContinuousOn.tendstoUniformlyproof · cited by 0
- Continuous.tendstoUniformlyproof · cited by 0