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Theorems · Theorem · general topology

Metric.diam_singleton

∀ {α : Type u} {x : α} [inst : PseudoMetricSpace α], Metric.diam {x} = 0

A singleton has zero diameter

Defined in
Mathlib.Topology.MetricSpace.Bounded
Cited by
3 results in Mathlib
Foundations
Depth 152 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.

Metric.diam_one · cited by 0Metric.diam_oneMetric.diam_zero · cited by 0Metric.diam_zerodiam_stdSimplex_of_subsingleton · cited by 0diam_stdSimplex_of_subsin…Set · cited by 53352SetReal · cited by 25697RealPseudoMetricSpace · cited by 1550PseudoMetricSpaceMetric.diam · cited by 74Metric.diamSet.subsingleton_singleton · cited by 36Set.subsingleton_singletonMetric.diam_subsingleton · cited by 2Metric.diam_subsingletonMetric.diam_singletonCITED BYCITES

Cites6

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.