Theorems · Theorem · general topology
Metric.isBounded_range_iff
∀ {α : Type u} {β : Type v} [inst : PseudoMetricSpace α] {f : β → α},
Bornology.IsBounded (Set.range f) ↔ ∃ C, ∀ (x y : β), dist (f x) (f y) ≤ CCharacterization of the boundedness of the range of a function
- Defined in
- Mathlib.Topology.MetricSpace.Bounded
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 97 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- PseudoMetricSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- Set.rangestatement and proof · cited by 4,705
- PseudoMetricSpacestatement and proof · cited by 1,550
- Dist.diststatement and proof · cited by 1,539
- Bornology.IsBoundedstatement · cited by 293
- Metric.isBounded_iffproof · cited by 15
Cited by8
Results whose statement or proof uses this declaration.
- BoundedContinuousFunction.isBounded_rangeproof · cited by 7
- MeasureTheory.MemLp.exists_boundedContinuous_eLpNorm_sub_leproof · cited by 3
- tendsto_birkhoffAverage_apply_sub_birkhoffAverageproof · cited by 2
- ContinuousMap.exists_extension_forall_mem_of_isClosedEmbeddingproof · cited by 1
- ZeroAtInftyContinuousMap.isBounded_rangeproof · cited by 1
- mul_bounded_of_bounded_of_boundedproof · cited by 0
- add_bounded_of_bounded_of_boundedproof · cited by 0
- sub_bounded_of_bounded_of_boundedproof · cited by 0