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Theorems · Theorem · general topology

Metric.infDist_nonneg

∀ {α : Type u} [inst : PseudoMetricSpace α] {s : Set α} {x : α}, 0 ≤ Metric.infDist x s

The minimal distance is always nonnegative

Defined in
Mathlib.Topology.MetricSpace.HausdorffDistance
Cited by
3 results in Mathlib
Foundations
Depth 149 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
PseudoMetricSpace
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

IsClosed.notMem_iff_infDist_pos · cited by 4IsClosed.notMem_iff_infDi…riesz_lemma · cited by 2riesz_lemmaMetric.infDist_pos_iff_notMem_closure · cited by 0Metric.infDist_pos_iff_no…Set · cited by 53352SetReal · cited by 25697RealPseudoMetricSpace · cited by 1550PseudoMetricSpaceENNReal.toReal_nonneg · cited by 86ENNReal.toReal_nonnegMetric.infDist · cited by 68Metric.infDistMetric.infDist_nonnegCITED BYCITES

Cites5

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by3

Results whose statement or proof uses this declaration.