Theorems · Theorem · general topology
EMetric.nonempty_of_hausdorffEdist_ne_top
Deprecated since 2026-01-08Use Metric.nonempty_of_hausdorffEDist_ne_top instead.
∀ {α : Type u} [inst : PseudoEMetricSpace α] {s t : Set α}, s.Nonempty → Metric.hausdorffEDist s t ≠ ⊤ → t.NonemptyAlias of Metric.nonempty_of_hausdorffEDist_ne_top.
If a set is at finite Hausdorff edistance of a nonempty set, it is nonempty.
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- 0 results in Mathlib
- Foundations
- Depth 156 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- PseudoEMetricSpace
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Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- ENNRealstatement · cited by 9,879
- Top.topstatement · cited by 9,680
- Set.Nonemptystatement · cited by 2,627
- PseudoEMetricSpacestatement · cited by 1,536
- Metric.hausdorffEDiststatement · cited by 76
- Metric.nonempty_of_hausdorffEDist_ne_topproof · cited by 3
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