Theorems · Theorem · general topology
Metric.mem_ball
∀ {α : Type u} [inst : PseudoMetricSpace α] {x y : α} {ε : ℝ}, y ∈ Metric.ball x ε ↔ dist y x < ε- Defined in
- Mathlib.Topology.MetricSpace.Pseudo.Defs
- Cited by
- 47 results in Mathlib
- Foundations
- Depth 91 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.
Cites5
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.diststatement · cited by 1,539
- Metric.ballstatement · cited by 735
Cited by47
Results whose statement or proof uses this declaration.
- Metric.mem_ball_selfproof · cited by 40
- mem_ball_zero_iffproof · cited by 22
- closure_ballproof · cited by 20
- mem_ball_iff_normproof · cited by 20
- Metric.ball_subset_ballproof · cited by 18
- Real.ball_eq_Iooproof · cited by 11
- Metric.mem_ball'proof · cited by 10
- ball_zero_eqproof · cited by 8
- ContinuousLinearMap.isOpenMapproof · cited by 6
- multipliable_one_add_of_summableproof · cited by 4
- MeasureTheory.hasSum_integral_measureproof · cited by 4
- preimage_add_ballproof · cited by 3