# mathlibdocumentation

topology.metric_space.closeds

# Closed subsets #

This file defines the metric and emetric space structure on the types of closed subsets and nonempty compact subsets of a metric or emetric space.

The Hausdorff distance induces an emetric space structure on the type of closed subsets of an emetric space, called closeds. Its completeness, resp. compactness, resp. second-countability, follow from the corresponding properties of the original space.

In a metric space, the type of nonempty compact subsets (called nonempty_compacts) also inherits a metric space structure from the Hausdorff distance, as the Hausdorff edistance is always finite in this context.

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noncomputable def emetric.closeds.emetric_space {α : Type u}  :

In emetric spaces, the Hausdorff edistance defines an emetric space structure on the type of closed subsets

Equations
theorem emetric.continuous_inf_edist_Hausdorff_edist {α : Type u}  :
continuous (λ (p : , p.snd.val)

The edistance to a closed set depends continuously on the point and the set

theorem emetric.is_closed_subsets_of_is_closed {α : Type u} {s : set α} (hs : is_closed s) :
is_closed {t : | t.val s}

Subsets of a given closed subset form a closed set

theorem emetric.closeds.edist_eq {α : Type u} {s t : topological_space.closeds α} :
t =

By definition, the edistance on closeds α is given by the Hausdorff edistance

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def emetric.closeds.complete_space {α : Type u}  :

In a complete space, the type of closed subsets is complete for the Hausdorff edistance.

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def emetric.closeds.compact_space {α : Type u}  :

In a compact space, the type of closed subsets is compact.

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noncomputable def emetric.nonempty_compacts.emetric_space {α : Type u}  :

In an emetric space, the type of non-empty compact subsets is an emetric space, where the edistance is the Hausdorff edistance

Equations

nonempty_compacts.to_closeds is a uniform embedding (as it is an isometry)

The range of nonempty_compacts.to_closeds is closed in a complete space

@[protected, instance]

In a complete space, the type of nonempty compact subsets is complete. This follows from the same statement for closed subsets

@[protected, instance]

In a compact space, the type of nonempty compact subsets is compact. This follows from the same statement for closed subsets

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In a second countable space, the type of nonempty compact subsets is second countable

@[protected, instance]
noncomputable def metric.nonempty_compacts.metric_space {α : Type u} [metric_space α] :

nonempty_compacts α inherits a metric space structure, as the Hausdorff edistance between two such sets is finite.

Equations
theorem metric.nonempty_compacts.dist_eq {α : Type u} [metric_space α]  :
dist x y =

The distance on nonempty_compacts α is the Hausdorff distance, by construction

theorem metric.lipschitz_inf_dist_set {α : Type u} [metric_space α] (x : α) :
(λ (s : , s.val)
theorem metric.lipschitz_inf_dist {α : Type u} [metric_space α] :
(λ (p : , p.snd.val)