Theorems · Theorem · general topology
Metric.isOpen_iff
∀ {α : Type u} [inst : PseudoMetricSpace α] {s : Set α}, IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s- Defined in
- Mathlib.Topology.MetricSpace.Pseudo.Defs
- Cited by
- 22 results in Mathlib
- Foundations
- Depth 117 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 and proof · cited by 53,352
- Realstatement and proof · cited by 25,697
- IsOpenstatement · cited by 2,400
- PseudoMetricSpacestatement and proof · cited by 1,550
- Metric.ballstatement and proof · cited by 735
Cited by22
Results whose statement or proof uses this declaration.
- Metric.isOpen_ballproof · cited by 63
- Units.isOpenproof · cited by 6
- ContinuousLinearMap.isOpenMapproof · cited by 6
- PeriodPair.summable_weierstrassPExceptSummandproof · cited by 3
- IsOpen.isOpen_inter_preimage_of_fderiv_eq_zeroproof · cited by 3
- IsUltrametricDist.isOpen_closedBallproof · cited by 3
- AnalyticAt.exists_ball_analyticOnNhdproof · cited by 2
- ZLattice.exists_forall_abs_repr_le_normproof · cited by 2
- AnalyticAt.eventually_constant_or_nhds_le_map_nhdsproof · cited by 1
- tendsto_integral_comp_smul_smul_of_integrableproof · cited by 1
- Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zeroproof · cited by 1
- MeasureTheory.hasFDerivAt_convolution_right_with_paramproof · cited by 1