Theorems · Theorem · general topology
Metric.mem_closedBall
∀ {α : Type u} [inst : PseudoMetricSpace α] {x y : α} {ε : ℝ}, y ∈ Metric.closedBall x ε ↔ dist y x ≤ ε- Defined in
- Mathlib.Topology.MetricSpace.Pseudo.Defs
- Cited by
- 42 results in Mathlib
- Foundations
- Depth 96 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.closedBallstatement · cited by 704
Cited by42
Results whose statement or proof uses this declaration.
- Metric.ball_subset_closedBallproof · cited by 46
- closure_ballproof · cited by 20
- mem_closedBall_zero_iffproof · cited by 16
- Real.closedBall_eq_Iccproof · cited by 13
- mem_closedBall_iff_normproof · cited by 12
- Metric.mem_closedBall_selfproof · cited by 12
- Metric.mem_closedBall'proof · cited by 11
- interior_closedBallproof · cited by 8
- IsCompact.add_closedBall_zeroproof · cited by 5
- Metric.closedBall_zero'proof · cited by 4
- Complex.norm_eq_norm_of_isMaxOn_of_ball_subsetproof · cited by 3
- ODE.FunSpace.mem_closedBallproof · cited by 3