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Theorems · Theorem · general topology

Metric.infDist_closure

∀ {α : Type u} [inst : PseudoMetricSpace α] {s : Set α} {x : α}, Metric.infDist x (closure s) = Metric.infDist x s

The minimal distances to a set and its closure coincide.

Defined in
Mathlib.Topology.MetricSpace.HausdorffDistance
Cited by
2 results in Mathlib
Foundations
Depth 150 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
PseudoMetricSpace
Mathlib docsSourceLoogleLeanSearch

Around this declaration

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

Metric.exists_mem_closure_infDist_eq_dist · cited by 1Metric.exists_mem_closure…Metric.infDist_zero_of_mem_closure · cited by 1Metric.infDist_zero_of_me…Set · cited by 53352SetReal · cited by 25697RealPseudoMetricSpace · cited by 1550PseudoMetricSpaceclosure · cited by 1254closureENNReal.toReal · cited by 859ENNReal.toRealMetric.infEDist · cited by 147Metric.infEDistMetric.infDist · cited by 68Metric.infDistMetric.infEDist_closure · cited by 8Metric.infEDist_closureMetric.infDist_closureCITED BYCITES

Cites8

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Cited by2

Results whose statement or proof uses this declaration.