Theorems · Theorem · general topology
Metric.isBounded_iff_subset_ball
∀ {α : Type u} {s : Set α} [inst : PseudoMetricSpace α] (c : α), Bornology.IsBounded s ↔ ∃ r, s ⊆ Metric.ball c r- Defined in
- Mathlib.Topology.MetricSpace.Bounded
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 112 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.
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
- Realstatement and proof · cited by 25,697
- PseudoMetricSpacestatement and proof · cited by 1,550
- Metric.ballstatement and proof · cited by 735
- Bornology.IsBoundedstatement and proof · cited by 293
- Bornology.IsBounded.subsetproof · cited by 45
- Metric.isBounded_ballproof · cited by 15
- Bornology.IsBounded.subset_ballproof · cited by 2
Cited by3
Results whose statement or proof uses this declaration.
- Metric.hasBasis_cobounded_compl_ballproof · cited by 2
- ConvexOn.exists_lipschitzOnWith_of_isBoundedproof · cited by 2
- AddMonoidHom.continuous_of_isBounded_nhds_zeroproof · cited by 1