# mathlibdocumentation

topology.metric_space.kuratowski

# The Kuratowski embedding #

Any separable metric space can be embedded isometrically in ℓ^∞(ℝ).

def ℓ_infty_ℝ  :
Type

The space of bounded sequences, with its sup norm

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### Any separable metric space can be embedded isometrically in ℓ^∞(ℝ) #

def Kuratowski_embedding.embedding_of_subset {α : Type u} [metric_space α] (x : → α) (a : α) :

A metric space can be embedded in l^∞(ℝ) via the distances to points in a fixed countable set, if this set is dense. This map is given in Kuratowski_embedding, without density assumptions.

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theorem Kuratowski_embedding.embedding_of_subset_coe {α : Type u} {n : } [metric_space α] (x : → α) (a : α) :
= dist a (x n) - dist (x 0) (x n)
theorem Kuratowski_embedding.embedding_of_subset_dist_le {α : Type u} [metric_space α] (x : → α) (a b : α) :

The embedding map is always a semi-contraction.

theorem Kuratowski_embedding.embedding_of_subset_isometry {α : Type u} [metric_space α] (x : → α) (H : dense_range x) :

When the reference set is dense, the embedding map is an isometry on its image.

theorem Kuratowski_embedding.exists_isometric_embedding (α : Type u) [metric_space α]  :
∃ (f : α → ℓ_infty_ℝ),

Every separable metric space embeds isometrically in ℓ_infty_ℝ.

def Kuratowski_embedding (α : Type u) [metric_space α]  :

The Kuratowski embedding is an isometric embedding of a separable metric space in ℓ^∞(ℝ).

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theorem Kuratowski_embedding.isometry (α : Type u) [metric_space α]  :

The Kuratowski embedding is an isometry.

Version of the Kuratowski embedding for nonempty compacts

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