Theorems · Theorem · general topology
LipschitzWith.isBounded_image
∀ {α : Type u} {β : Type v} [inst : PseudoMetricSpace α] [inst_1 : PseudoMetricSpace β] {K : NNReal} {f : α → β},
LipschitzWith K f → ∀ {s : Set α}, Bornology.IsBounded s → Bornology.IsBounded (f '' s)The image of a bounded set under a Lipschitz map is bounded.
- Defined in
- Mathlib.Topology.MetricSpace.Lipschitz
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 153 from the axioms · uses propext, Classical.choice, Quot.sound
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
- Set.imagestatement · cited by 5,609
- NNRealstatement and proof · cited by 4,310
- PseudoMetricSpacestatement and proof · cited by 1,550
- LipschitzWithstatement and proof · cited by 316
- Bornology.IsBoundedstatement and proof · cited by 293
- LipschitzWith.toLocallyBoundedMapproof · cited by 3
- Bornology.IsBounded.imageproof · cited by 2
Cited by6
Results whose statement or proof uses this declaration.
- Bornology.IsBounded.smulproof · cited by 1
- ContinuousLinearMap.isCompact_closure_image_coe_of_boundedproof · cited by 1
- Bornology.IsBounded.vaddproof · cited by 1
- Bornology.IsBounded.smul₀proof · cited by 0
- NormedSpace.isCompact_closure_of_isBoundedproof · cited by 0
- boundedSub_of_lipschitzWith_subproof · cited by 0