File: Mathlib/Topology/MetricSpace/Closeds.lean

full_name: EMetric.Closeds.emetricSpace
code: /-- In emetric spaces, the Hausdorff edistance defines an emetric space structure
on the type of closed subsets -/
instance Closeds.emetricSpace : EMetricSpace (Closeds α) 
goal: EMetricSpace (TopologicalSpace.Closeds α)

full_name: EMetric.continuous_infEdist_hausdorffEdist
code: /-- The edistance to a closed set depends continuously on the point and the set -/
theorem continuous_infEdist_hausdorffEdist :
    Continuous fun p : α × Closeds α => infEdist p.1 p.2 
goal: Continuous fun p => EMetric.infEdist p.1 ↑p.2

full_name: EMetric.isClosed_subsets_of_isClosed
code: /-- Subsets of a given closed subset form a closed set -/
theorem isClosed_subsets_of_isClosed (hs : IsClosed s) :
    IsClosed { t : Closeds α | (t : Set α) ⊆ s } 
goal: IsClosed {t | ↑t ⊆ s}

full_name: EMetric.Closeds.edist_eq
code: /-- By definition, the edistance on `Closeds α` is given by the Hausdorff edistance -/
theorem Closeds.edist_eq {s t : Closeds α} : edist s t = hausdorffEdist (s : Set α) t 
goal: edist s t = EMetric.hausdorffEdist ↑s ↑t

full_name: EMetric.Closeds.completeSpace
code: /-- In a complete space, the type of closed subsets is complete for the
Hausdorff edistance. -/
instance Closeds.completeSpace [CompleteSpace α] : CompleteSpace (Closeds α) 
goal: CompleteSpace (TopologicalSpace.Closeds α)

full_name: EMetric.Closeds.compactSpace
code: /-- In a compact space, the type of closed subsets is compact. -/
instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) 
goal: CompactSpace (TopologicalSpace.Closeds α)

full_name: EMetric.NonemptyCompacts.emetricSpace
code: /-- In an emetric space, the type of non-empty compact subsets is an emetric space,
where the edistance is the Hausdorff edistance -/
instance NonemptyCompacts.emetricSpace : EMetricSpace (NonemptyCompacts α) 
goal: EMetricSpace (TopologicalSpace.NonemptyCompacts α)

full_name: EMetric.NonemptyCompacts.ToCloseds.uniformEmbedding
code: /-- `NonemptyCompacts.toCloseds` is a uniform embedding (as it is an isometry) -/
theorem NonemptyCompacts.ToCloseds.uniformEmbedding :
    UniformEmbedding (@NonemptyCompacts.toCloseds α _ _) 
goal: UniformEmbedding TopologicalSpace.NonemptyCompacts.toCloseds

full_name: EMetric.NonemptyCompacts.isClosed_in_closeds
code: /-- The range of `NonemptyCompacts.toCloseds` is closed in a complete space -/
theorem NonemptyCompacts.isClosed_in_closeds [CompleteSpace α] :
    IsClosed (range <| @NonemptyCompacts.toCloseds α _ _) 
goal: IsClosed (Set.range TopologicalSpace.NonemptyCompacts.toCloseds)

full_name: EMetric.NonemptyCompacts.completeSpace
code: /-- In a complete space, the type of nonempty compact subsets is complete. This follows
from the same statement for closed subsets -/
instance NonemptyCompacts.completeSpace [CompleteSpace α] : CompleteSpace (NonemptyCompacts α) 
goal: CompleteSpace (TopologicalSpace.NonemptyCompacts α)

full_name: EMetric.NonemptyCompacts.compactSpace
code: /-- In a compact space, the type of nonempty compact subsets is compact. This follows from
the same statement for closed subsets -/
instance NonemptyCompacts.compactSpace [CompactSpace α] : CompactSpace (NonemptyCompacts α) 
goal: CompactSpace (TopologicalSpace.NonemptyCompacts α)

full_name: EMetric.NonemptyCompacts.secondCountableTopology
code: /-- In a second countable space, the type of nonempty compact subsets is second countable -/
instance NonemptyCompacts.secondCountableTopology [SecondCountableTopology α] :
    SecondCountableTopology (NonemptyCompacts α) 
goal: SecondCountableTopology (TopologicalSpace.NonemptyCompacts α)

full_name: Metric.NonemptyCompacts.metricSpace
code: /-- `NonemptyCompacts α` inherits a metric space structure, as the Hausdorff
edistance between two such sets is finite. -/
instance NonemptyCompacts.metricSpace : MetricSpace (NonemptyCompacts α) 
goal: MetricSpace (TopologicalSpace.NonemptyCompacts α)

full_name: Metric.NonemptyCompacts.dist_eq
code: /-- The distance on `NonemptyCompacts α` is the Hausdorff distance, by construction -/
theorem NonemptyCompacts.dist_eq {x y : NonemptyCompacts α} :
    dist x y = hausdorffDist (x : Set α) y 
goal: dist x y = Metric.hausdorffDist ↑x ↑y

full_name: Metric.lipschitz_infDist_set
code: theorem lipschitz_infDist_set (x : α) : LipschitzWith 1 fun s : NonemptyCompacts α => infDist x s 
goal: LipschitzWith 1 fun s => Metric.infDist x ↑s

full_name: Metric.lipschitz_infDist
code: theorem lipschitz_infDist : LipschitzWith 2 fun p : α × NonemptyCompacts α => infDist p.1 p.2 
goal: LipschitzWith 2 fun p => Metric.infDist p.1 ↑p.2

full_name: Metric.uniformContinuous_infDist_Hausdorff_dist
code: theorem uniformContinuous_infDist_Hausdorff_dist :
    UniformContinuous fun p : α × NonemptyCompacts α => infDist p.1 p.2 
goal: UniformContinuous fun p => Metric.infDist p.1 ↑p.2