Theorems · Definition · general topology

MetricSpace.replaceTopology

{γ : Type u_3} →
  [U : TopologicalSpace γ] →
    (m : MetricSpace γ) → U = PseudoMetricSpace.toUniformSpace.toTopologicalSpace → MetricSpace γ

Build a new metric space from an old one where the bundled topological structure is provably (but typically non-definitionaly) equal to some given topological structure. See Note [forgetful inheritance]. See Note [reducible non-instances].

Defined in: Mathlib.Topology.MetricSpace.Defs
Cited by: 1 results in Mathlib
Foundations: Depth 120 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: TopologicalSpace

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

TopologicalSpacestatement and proof · cited by 24,529
MetricSpacestatement and proof · cited by 1,684
MetricSpace.replaceUniformityproof · cited by 1

Cited by3

Results whose statement or proof uses this declaration.

Topology.IsEmbedding.comapMetricSpaceproof · cited by 2
TopologicalSpace.completelyMetrizableMetricproof · cited by 1
MetricSpace.replaceTopology_eqstatement · cited by 1