Theorems · Theorem · general topology
Metric.mem_nhds_iff
∀ {α : Type u} [inst : PseudoMetricSpace α] {x : α} {s : Set α}, s ∈ nhds x ↔ ∃ ε > 0, Metric.ball x ε ⊆ s- Defined in
- Mathlib.Topology.MetricSpace.Pseudo.Defs
- Cited by
- 26 results in Mathlib
- Foundations
- Depth 115 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.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Realstatement · cited by 25,697
- Filterstatement · cited by 8,121
- nhdsstatement · cited by 5,554
- PseudoMetricSpacestatement and proof · cited by 1,550
- Metric.ballstatement · cited by 735
- Filter.HasBasis.mem_iffproof · cited by 193
- Metric.nhds_basis_ballproof · cited by 41
Cited by26
Results whose statement or proof uses this declaration.
- Metric.eventually_nhds_iff_ballproof · cited by 10
- hasFDerivAt_integral_of_dominated_of_fderiv_leproof · cited by 5
- Metric.eventually_nhds_iffproof · cited by 5
- IsOpen.analyticOn_iff_analyticOnNhdproof · cited by 5
- hasDerivAt_integral_of_dominated_loc_of_deriv_leproof · cited by 4
- ConvexOn.continuousOn_tfaeproof · cited by 3
- Irrational.eventually_forall_le_dist_cast_divproof · cited by 2
- isMIntegralCurveAt_iff'proof · cited by 2
- setOfPred_riemannianEDist_lt_subset_nhdsproof · cited by 2
- exists_mem_interior_convexHull_affineBasisproof · cited by 2
- hasStrictFDerivAt_of_hasFDerivAt_of_continuousAtproof · cited by 2
- MeasureTheory.measurablySeparable_range_of_disjointproof · cited by 1