Theorems · Theorem · general topology
Metric.isBounded_iff
∀ {α : Type u} [inst : PseudoMetricSpace α] {s : Set α},
Bornology.IsBounded s ↔ ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C- Defined in
- Mathlib.Topology.MetricSpace.Pseudo.Defs
- Cited by
- 15 results in Mathlib
- Foundations
- Depth 96 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.
Cites11
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
- Realstatement and proof · cited by 25,697
- Compl.complproof · cited by 2,925
- PseudoMetricSpacestatement and proof · cited by 1,550
- Dist.diststatement and proof · cited by 1,539
- Bornology.IsBoundedstatement · cited by 293
- compl_complproof · cited by 229
- Set.mem_ofPred_eqproof · cited by 122
- Bornology.isBounded_defproof · cited by 9
- Filter.mem_setsproof · cited by 3
- PseudoMetricSpace.cobounded_setsproof · cited by 1
Cited by15
Results whose statement or proof uses this declaration.
- Metric.isBounded_closedBallproof · cited by 19
- Metric.isBounded_iff_subset_closedBallproof · cited by 10
- Metric.isBounded_range_iffproof · cited by 8
- Metric.isBounded_iff_ediam_ne_topproof · cited by 5
- TotallyBounded.isVonNBoundedproof · cited by 4
- BoundedContinuousFunction.dist_set_existsproof · cited by 3
- Metric.isBounded_iff_eventuallyproof · cited by 2
- NumberField.mixedEmbedding.convexBodySum_isBoundedproof · cited by 1
- Metric.isBounded_image_iffproof · cited by 1
- Metric.isBounded_closure_of_isBoundedproof · cited by 1
- Metric.isBounded_iff_exists_geproof · cited by 1
- Metric.boundedSpace_iffproof · cited by 1