File: Mathlib/Topology/Metrizable/Basic.lean

full_name: TopologicalSpace.PseudoMetrizableSpace.exists_pseudo_metric
code: /-- A topological space is *pseudo metrizable* if there exists a pseudo metric space structure
compatible with the topology. To endow such a space with a compatible distance, use
`letI : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X`. -/
class PseudoMetrizableSpace (X : Type*) [t : TopologicalSpace X] : Prop where
  exists_pseudo_metric : ∃ m : PseudoMetricSpace X, m.toUniformSpace.toTopologicalSpace = t
goal: ∃ m, UniformSpace.toTopologicalSpace = t

full_name: TopologicalSpace.PseudoMetrizableSpace
code: /-- A topological space is *pseudo metrizable* if there exists a pseudo metric space structure
compatible with the topology. To endow such a space with a compatible distance, use
`letI : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X`. -/
class PseudoMetrizableSpace (X : Type*) [t : TopologicalSpace X] : Prop where
  exists_pseudo_metric : ∃ m : PseudoMetricSpace X, m.toUniformSpace.toTopologicalSpace = t
goal: Prop

full_name: PseudoMetricSpace.toPseudoMetrizableSpace
code: instance (priority := 100) _root_.PseudoMetricSpace.toPseudoMetrizableSpace {X : Type*}
    [m : PseudoMetricSpace X] : PseudoMetrizableSpace X 
goal: TopologicalSpace.PseudoMetrizableSpace X

full_name: TopologicalSpace.pseudoMetrizableSpacePseudoMetric
code: /-- Construct on a metrizable space a metric compatible with the topology. -/
noncomputable def pseudoMetrizableSpacePseudoMetric (X : Type*) [TopologicalSpace X]
    [h : PseudoMetrizableSpace X] : PseudoMetricSpace X 
goal: PseudoMetricSpace X

full_name: TopologicalSpace.pseudoMetrizableSpace_prod
code: instance pseudoMetrizableSpace_prod [PseudoMetrizableSpace X] [PseudoMetrizableSpace Y] :
    PseudoMetrizableSpace (X × Y) 
goal: TopologicalSpace.PseudoMetrizableSpace (X × Y)

full_name: Inducing.pseudoMetrizableSpace
code: /-- Given an inducing map of a topological space into a pseudo metrizable space, the source space
is also pseudo metrizable. -/
theorem _root_.Inducing.pseudoMetrizableSpace [PseudoMetrizableSpace Y] {f : X → Y}
    (hf : Inducing f) : PseudoMetrizableSpace X 
goal: TopologicalSpace.PseudoMetrizableSpace X

full_name: TopologicalSpace.PseudoMetrizableSpace.firstCountableTopology
code: /-- Every pseudo-metrizable space is first countable. -/
instance (priority := 100) PseudoMetrizableSpace.firstCountableTopology
    [h : PseudoMetrizableSpace X] : FirstCountableTopology X 
goal: FirstCountableTopology X

full_name: TopologicalSpace.PseudoMetrizableSpace.subtype
code: instance PseudoMetrizableSpace.subtype [PseudoMetrizableSpace X] (s : Set X) :
    PseudoMetrizableSpace s 
goal: TopologicalSpace.PseudoMetrizableSpace ↑s

full_name: TopologicalSpace.pseudoMetrizableSpace_pi
code: instance pseudoMetrizableSpace_pi [∀ i, PseudoMetrizableSpace (π i)] :
    PseudoMetrizableSpace (∀ i, π i) 
goal: TopologicalSpace.PseudoMetrizableSpace ((i : ι) → π i)

full_name: TopologicalSpace.MetrizableSpace
code: /-- A topological space is metrizable if there exists a metric space structure compatible with the
topology. To endow such a space with a compatible distance, use
`letI : MetricSpace X := TopologicalSpace.metrizableSpaceMetric X`. -/
class MetrizableSpace (X : Type*) [t : TopologicalSpace X] : Prop where
  exists_metric : ∃ m : MetricSpace X, m.toUniformSpace.toTopologicalSpace = t
goal: Prop

full_name: TopologicalSpace.MetrizableSpace.exists_metric
code: /-- A topological space is metrizable if there exists a metric space structure compatible with the
topology. To endow such a space with a compatible distance, use
`letI : MetricSpace X := TopologicalSpace.metrizableSpaceMetric X`. -/
class MetrizableSpace (X : Type*) [t : TopologicalSpace X] : Prop where
  exists_metric : ∃ m : MetricSpace X, m.toUniformSpace.toTopologicalSpace = t
goal: ∃ m, UniformSpace.toTopologicalSpace = t

full_name: MetricSpace.toMetrizableSpace
code: instance (priority := 100) _root_.MetricSpace.toMetrizableSpace {X : Type*} [m : MetricSpace X] :
    MetrizableSpace X 
goal: TopologicalSpace.MetrizableSpace X

full_name: TopologicalSpace.MetrizableSpace.toPseudoMetrizableSpace
code: instance (priority := 100) MetrizableSpace.toPseudoMetrizableSpace [h : MetrizableSpace X] :
    PseudoMetrizableSpace X 
goal: TopologicalSpace.PseudoMetrizableSpace X

full_name: TopologicalSpace.metrizableSpaceMetric
code: /-- Construct on a metrizable space a metric compatible with the topology. -/
noncomputable def metrizableSpaceMetric (X : Type*) [TopologicalSpace X] [h : MetrizableSpace X] :
    MetricSpace X 
goal: MetricSpace X

full_name: TopologicalSpace.t2Space_of_metrizableSpace
code: instance (priority := 100) t2Space_of_metrizableSpace [MetrizableSpace X] : T2Space X 
goal: T2Space X

full_name: TopologicalSpace.metrizableSpace_prod
code: instance metrizableSpace_prod [MetrizableSpace X] [MetrizableSpace Y] : MetrizableSpace (X × Y) 
goal: TopologicalSpace.MetrizableSpace (X × Y)

full_name: Embedding.metrizableSpace
code: /-- Given an embedding of a topological space into a metrizable space, the source space is also
metrizable. -/
theorem _root_.Embedding.metrizableSpace [MetrizableSpace Y] {f : X → Y} (hf : Embedding f) :
    MetrizableSpace X 
goal: TopologicalSpace.MetrizableSpace X

full_name: TopologicalSpace.MetrizableSpace.subtype
code: instance MetrizableSpace.subtype [MetrizableSpace X] (s : Set X) : MetrizableSpace s 
goal: TopologicalSpace.MetrizableSpace ↑s

full_name: TopologicalSpace.metrizableSpace_pi
code: instance metrizableSpace_pi [∀ i, MetrizableSpace (π i)] : MetrizableSpace (∀ i, π i) 
goal: TopologicalSpace.MetrizableSpace ((i : ι) → π i)

full_name: TopologicalSpace.IsSeparable.secondCountableTopology
code: theorem IsSeparable.secondCountableTopology [PseudoMetrizableSpace X] {s : Set X}
    (hs : IsSeparable s) : SecondCountableTopology s 
goal: SecondCountableTopology ↑s

full_name: TopologicalSpace.instSecondCountableTopologyOfCompactSpaceOfMetrizableSpace
code: instance (X : Type*) [TopologicalSpace X] [c : CompactSpace X] [MetrizableSpace X] :
    SecondCountableTopology X 
goal: SecondCountableTopology X