Theorems · Theorem · general topology
Metric.closedBall_subset_ball
∀ {α : Type u} [inst : PseudoMetricSpace α] {x : α} {ε₁ ε₂ : ℝ}, ε₁ < ε₂ → Metric.closedBall x ε₁ ⊆ Metric.ball x ε₂- Defined in
- Mathlib.Topology.MetricSpace.Pseudo.Defs
- Cited by
- 20 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.
Cites7
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
- Metric.ballstatement · cited by 735
- Metric.closedBallstatement · cited by 704
- lt_of_le_of_ltproof · cited by 432
Cited by20
Results whose statement or proof uses this declaration.
- Bornology.IsBounded.subset_ball_ltproof · cited by 4
- exists_pos_lt_subset_ballproof · cited by 4
- Complex.hasSum_taylorSeries_on_ballproof · cited by 3
- AddCircle.closedBall_ae_eq_ballproof · cited by 2
- BoxIntegral.integrable_of_bounded_and_ae_continuousWithinAtproof · cited by 2
- MeasureTheory.Measure.addHaar_unitClosedBall_eq_addHaar_unitBallproof · cited by 2
- exists_mem_interior_convexHull_affineBasisproof · cited by 2
- Metric.disjoint_closedBall_of_lt_infDistproof · cited by 1
- sub_isBigO_norm_rpow_add_one_of_fderivproof · cited by 1
- Complex.continuousOn_prod_circle_transform_functionproof · cited by 1
- gauge_closedBallproof · cited by 1