Theorems · Theorem · general topology
Metric.nonempty_of_hausdorffEDist_ne_top
∀ {α : Type u} [inst : PseudoEMetricSpace α] {s t : Set α}, s.Nonempty → Metric.hausdorffEDist s t ≠ ⊤ → t.NonemptyIf a set is at finite Hausdorff edistance of a nonempty set, it is nonempty.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 155 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- PseudoEMetricSpace
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
- ENNRealstatement · cited by 9,879
- Top.topstatement and proof · cited by 9,680
- Set.Nonemptystatement and proof · cited by 2,627
- PseudoEMetricSpacestatement and proof · cited by 1,536
- Set.eq_empty_or_nonemptyproof · cited by 248
- Metric.hausdorffEDiststatement and proof · cited by 76
- Metric.hausdorffEDist_emptyproof · cited by 3
Cited by3
Results whose statement or proof uses this declaration.
- Metric.empty_or_nonempty_of_hausdorffEDist_ne_topproof · cited by 1
- Metric.infDist_le_infDist_add_hausdorffDistproof · cited by 1
- EMetric.nonempty_of_hausdorffEdist_ne_topproof · cited by 0