Theorems · Theorem · general topology
IsClosed.exists_infDist_eq_dist
∀ {α : Type u} [inst : PseudoMetricSpace α] {s : Set α} [ProperSpace α],
IsClosed s → s.Nonempty → ∀ (x : α), ∃ y ∈ s, Metric.infDist x s = dist x y- Cited by
- 2 results in Mathlib
- Foundations
- Depth 153 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- PseudoMetricSpaceProperSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
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
- Set.Nonemptystatement and proof · cited by 2,627
- IsClosedstatement and proof · cited by 1,639
- le_rflproof · cited by 1,558
- PseudoMetricSpacestatement and proof · cited by 1,550
- Dist.diststatement and proof · cited by 1,539
- IsCompactproof · cited by 1,282
- Metric.closedBallproof · cited by 704
- ProperSpacestatement and proof · cited by 190
- Metric.infDiststatement and proof · cited by 68
- Metric.mem_closedBallproof · cited by 42
Cited by2
Results whose statement or proof uses this declaration.
- Metric.exists_mem_closure_infDist_eq_distproof · cited by 1
- Metric.cthickening_eq_biUnion_closedBallproof · cited by 1