Theorems · Theorem · general topology
Metric.isBounded_ball
∀ {α : Type u} {x : α} {r : ℝ} [inst : PseudoMetricSpace α], Bornology.IsBounded (Metric.ball x r)Open balls are bounded
- Defined in
- Mathlib.Topology.MetricSpace.Bounded
- Cited by
- 15 results in Mathlib
- Foundations
- Depth 104 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.
- Realstatement and proof · cited by 25,697
- PseudoMetricSpacestatement and proof · cited by 1,550
- Metric.ballstatement · cited by 735
- Bornology.IsBoundedstatement · cited by 293
- Metric.ball_subset_closedBallproof · cited by 46
- Bornology.IsBounded.subsetproof · cited by 45
- Metric.isBounded_closedBallproof · cited by 19
Cited by15
Results whose statement or proof uses this declaration.
- NormedSpace.isVonNBounded_iffproof · cited by 9
- Metric.disjoint_nhds_coboundedproof · cited by 7
- TotallyBounded.isBoundedproof · cited by 5
- NormedSpace.isVonNBounded_ballproof · cited by 4
- TendstoLocallyUniformlyOn.smul₀_of_isBoundedUnderproof · cited by 4
- Metric.isBounded_iff_subset_ballproof · cited by 3
- MeasureTheory.measurableSet_range_of_continuous_injectiveproof · cited by 1
- MeasureTheory.SeparableSpace.exists_measurable_partition_diam_leproof · cited by 1
- Metric.eq_countable_union_of_isBounded_of_isOpenproof · cited by 1
- Convex.addHaar_frontierproof · cited by 1
- MeasureTheory.measure_ball_ne_topproof · cited by 1
- AddMonoidHom.exists_nhds_isBoundedproof · cited by 1