Theorems · Theorem · general topology
Metric.mem_closedBall_self
∀ {α : Type u} [inst : PseudoMetricSpace α] {x : α} {ε : ℝ}, 0 ≤ ε → x ∈ Metric.closedBall x ε- Defined in
- Mathlib.Topology.MetricSpace.Pseudo.Defs
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 97 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
- Metric.closedBallstatement · cited by 704
- dist_selfproof · cited by 116
- Metric.mem_closedBallproof · cited by 42
Cited by12
Results whose statement or proof uses this declaration.
- Metric.nonempty_closedBallproof · cited by 13
- IsUnifLocDoublingMeasure.closedBall_mem_vitaliFamily_of_dist_le_mulproof · cited by 3
- exists_contDiff_tsupport_subsetproof · cited by 3
- DiffContOnCl.ball_subset_image_closedBallproof · cited by 1
- Liouville.exists_pos_real_of_irrational_rootproof · cited by 1
- segment_subset_closedBall_leftproof · cited by 1
- unique_topology_of_t2proof · cited by 1
- Besicovitch.exists_closedBall_covering_tsum_measure_leproof · cited by 1
- IsPicardLindelof.exists_eq_forall_mem_Icc_hasDerivWithinAt₀proof · cited by 0
- Vitali.exists_disjoint_subfamily_covering_enlargement_closedBallproof · cited by 0