Theorems · Theorem · general topology
IsClosed.notMem_iff_infDist_pos
∀ {α : Type u} [inst : PseudoMetricSpace α] {s : Set α} {x : α},
IsClosed s → s.Nonempty → (x ∉ s ↔ 0 < Metric.infDist x s)Given a closed set s, a point belongs to s iff its infimum distance to this set vanishes.
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 154 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.
Cites9
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
- Set.Nonemptystatement and proof · cited by 2,627
- IsClosedstatement and proof · cited by 1,639
- PseudoMetricSpacestatement and proof · cited by 1,550
- Metric.infDiststatement and proof · cited by 68
- LE.le.lt_iff_ne'proof · cited by 16
- Metric.infDist_nonnegproof · cited by 3
- IsClosed.mem_iff_infDist_zeroproof · cited by 3
Cited by4
Results whose statement or proof uses this declaration.
- PiNat.exists_disjoint_cylinderproof · cited by 2
- Besicovitch.exists_disjoint_closedBall_covering_ae_of_finiteMeasure_auxproof · cited by 1
- MeasureTheory.norm_resolvent_le_inv_infDist_supportproof · cited by 1
- MeasureTheory.hasDerivAt_resolventTransformproof · cited by 1