Theorems · Theorem · general topology
Metric.closedBall_subset_closedBall
∀ {α : Type u} [inst : PseudoMetricSpace α] {x : α} {ε₁ ε₂ : ℝ},
ε₁ ≤ ε₂ → Metric.closedBall x ε₁ ⊆ Metric.closedBall x ε₂- Defined in
- Mathlib.Topology.MetricSpace.Pseudo.Defs
- Cited by
- 18 results in Mathlib
- Foundations
- Depth 102 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.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- PseudoMetricSpacestatement and proof · cited by 1,550
- Dist.distproof · cited by 1,539
- le_transproof · cited by 985
- Metric.closedBallstatement · cited by 704
Cited by18
Results whose statement or proof uses this declaration.
- upperHemicontinuous_spectrumproof · cited by 4
- Besicovitch.tendsto_filterAtproof · cited by 4
- IsUnifLocDoublingMeasure.closedBall_mem_vitaliFamily_of_dist_le_mulproof · cited by 3
- eventually_closedBall_subsetproof · cited by 3
- IsPicardLindelof.shrinkproof · cited by 2
- BoxIntegral.TaggedPrepartition.IsSubordinate.mono'proof · cited by 2
- AnalyticAt.eventually_constant_or_nhds_le_map_nhds_auxproof · cited by 1
- BoxIntegral.hasIntegral_GP_pderivproof · cited by 1
- Vitali.exists_disjoint_covering_aeproof · cited by 1
- AddCircle.coe_real_preimage_closedBall_inter_eqproof · cited by 1