Theorems · Theorem · general topology
Metric.isBounded_iff_ediam_ne_top
∀ {α : Type u} {s : Set α} [inst : PseudoMetricSpace α], Bornology.IsBounded s ↔ Metric.ediam s ≠ ⊤Characterize the boundedness of a set in terms of the finiteness of its emetric.diameter.
- Defined in
- Mathlib.Topology.MetricSpace.Bounded
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 152 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.
Cites14
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
- Realproof · cited by 25,697
- ENNRealstatement · cited by 9,879
- Top.topstatement and proof · cited by 9,680
- PseudoMetricSpacestatement and proof · cited by 1,550
- Dist.distproof · cited by 1,539
- Bornology.IsBoundedstatement · cited by 293
- Metric.ediamstatement and proof · cited by 159
- Metric.diamproof · cited by 74
- ne_top_of_le_ne_topproof · cited by 68
- ENNReal.ofReal_ne_topproof · cited by 34
- Metric.isBounded_iffproof · cited by 15
Cited by5
Results whose statement or proof uses this declaration.
- AntilipschitzWith.isBounded_preimageproof · cited by 6
- Bornology.IsBounded.ediam_ne_topproof · cited by 5
- Metric.ediam_eq_top_iff_unboundedproof · cited by 1
- Metric.ediam_univ_eq_top_iff_noncompactproof · cited by 1
- LipschitzOnWith.isBounded_image2proof · cited by 0