Theorems · Theorem · general topology
Metric.sphere_eq_empty_of_subsingleton
∀ {α : Type u} [inst : PseudoMetricSpace α] {x : α} {ε : ℝ} [Subsingleton α], ε ≠ 0 → Metric.sphere x ε = ∅- Defined in
- Mathlib.Topology.MetricSpace.Pseudo.Defs
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 89 from the axioms · uses propext, Classical.choice, Quot.sound
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.spherestatement and proof · cited by 371
- Set.eq_empty_iff_forall_notMemproof · cited by 42
- Metric.ne_of_mem_sphereproof · cited by 5
Cited by3
Results whose statement or proof uses this declaration.
- Topology.RelCWComplex.cellFrontier_zero_eq_emptyproof · cited by 2
- ContinuousLinearMap.sSup_sphere_eq_nnnormproof · cited by 1
- EuclideanGeometry.Sphere.inter_orthRadius_eq_empty_iffproof · cited by 0