Theorems · Theorem · general topology
Metric.mem_ball_self
∀ {α : Type u} [inst : PseudoMetricSpace α] {x : α} {ε : ℝ}, 0 < ε → x ∈ Metric.ball x ε- Defined in
- Mathlib.Topology.MetricSpace.Pseudo.Defs
- Cited by
- 40 results in Mathlib
- Foundations
- Depth 92 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.ballstatement · cited by 735
- dist_selfproof · cited by 116
- Metric.mem_ballproof · cited by 47
Cited by40
Results whose statement or proof uses this declaration.
- Metric.ball_mem_nhdsproof · cited by 64
- Metric.nonempty_ballproof · cited by 13
- hasFDerivAt_integral_of_dominated_of_fderiv_leproof · cited by 5
- hasDerivAt_integral_of_dominated_loc_of_deriv_leproof · cited by 4
- IsOpen.isOpen_inter_preimage_of_fderiv_eq_zeroproof · cited by 3
- eventually_residual_liouvilleproof · cited by 2
- ModularForm.discriminant_qExpansion_coeff_oneproof · cited by 2
- hasFDerivWithinAt_closure_of_tendsto_fderivproof · cited by 2
- Complex.affine_of_mapsTo_ball_of_norm_dslope_eq_divproof · cited by 2
- Complex.cderiv_eq_derivproof · cited by 2
- uniformCauchySeqOn_ball_of_fderivproof · cited by 2
- ContDiffWithinAt.exists_lipschitzOnWithproof · cited by 2