Theorems · Theorem · general topology
Metric.isBounded_closedBall
∀ {α : Type u} {x : α} {r : ℝ} [inst : PseudoMetricSpace α], Bornology.IsBounded (Metric.closedBall x r)Closed balls are bounded
- Defined in
- Mathlib.Topology.MetricSpace.Bounded
- Cited by
- 19 results in Mathlib
- Foundations
- Depth 103 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.closedBallstatement and proof · cited by 704
- add_le_addproof · cited by 666
- Bornology.IsBoundedstatement · cited by 293
- dist_triangle_rightproof · cited by 16
- Metric.isBounded_iffproof · cited by 15
Cited by19
Results whose statement or proof uses this declaration.
- Metric.isBounded_ballproof · cited by 15
- Metric.isBounded_iff_subset_closedBallproof · cited by 10
- MeasureTheory.measure_closedBall_lt_topproof · cited by 6
- NormedSpace.isVonNBounded_closedBallproof · cited by 5
- ZLattice.rankproof · cited by 5
- spectrum.isBoundedproof · cited by 4
- WeakDual.isBounded_closedBallproof · cited by 3
- ApproximatesLinearOn.norm_fderiv_sub_leproof · cited by 3
- Metric.diam_sphere_eqproof · cited by 2
- MeasureTheory.Measure.addHaar_eq_zero_of_disjoint_translatesproof · cited by 1
- exists_nat_nat_continuous_surjective_of_completeSpaceproof · cited by 1
- ZLattice.summable_norm_sub_rpowproof · cited by 1