Theorems · Theorem · general topology
Bornology.IsBounded.ediam_ne_top
∀ {α : Type u} {s : Set α} [inst : PseudoMetricSpace α], Bornology.IsBounded s → Metric.ediam s ≠ ⊤Alias of the forward direction of Metric.isBounded_iff_ediam_ne_top.
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 153 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.
Cites7
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
- ENNRealstatement · cited by 9,879
- Top.topstatement · cited by 9,680
- PseudoMetricSpacestatement and proof · cited by 1,550
- Bornology.IsBoundedstatement · cited by 293
- Metric.ediamstatement · cited by 159
- Metric.isBounded_iff_ediam_ne_topproof · cited by 5
Cited by5
Results whose statement or proof uses this declaration.
- Metric.dist_le_diam_of_memproof · cited by 14
- Metric.diam_monoproof · cited by 7
- AntilipschitzWith.isBounded_preimageproof · cited by 6
- Metric.dist_le_infDist_add_diamproof · cited by 1
- LipschitzOnWith.isBounded_image2proof · cited by 0