Theorems · Theorem · general topology
Metric.closedBall_eq_empty
∀ {α : Type u} [inst : PseudoMetricSpace α] {x : α} {ε : ℝ}, Metric.closedBall x ε = ∅ ↔ ε < 0- Defined in
- Mathlib.Topology.MetricSpace.Pseudo.Defs
- Cited by
- 15 results in Mathlib
- Foundations
- Depth 113 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
- Metric.closedBallstatement · cited by 704
- not_leproof · cited by 328
- Set.not_nonempty_iff_eq_emptyproof · cited by 56
- Metric.nonempty_closedBallproof · cited by 13
Cited by15
Results whose statement or proof uses this declaration.
- MeasureTheory.Measure.addHaar_closedBall_eq_addHaar_ballproof · cited by 9
- interior_closedBallproof · cited by 8
- Metric.closedBall_of_negproof · cited by 3
- IsUltrametricDist.isOpen_closedBallproof · cited by 3
- Metric.subsingleton_closedBallproof · cited by 3
- IsUnifLocDoublingMeasure.eventually_measure_mul_le_scalingConstantOf_mulproof · cited by 2
- MeasureTheory.Measure.addHaar_sphere_of_ne_zeroproof · cited by 1
- NormedAddCommGroup.exists_norm_nsmul_leproof · cited by 1
- closedBall_rpow_sub_one_eq_empty_auxproof · cited by 1
- Metric.closedBall_subset_cthickening_singletonproof · cited by 1
- Nat.closedBall_eq_Iccproof · cited by 0
- closedBall_pi'proof · cited by 0