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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

Equations

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.

Equations
theorem Kuratowski_embedding.embedding_of_subset_coe {α : Type u} {n : } [metric_space α] (x : → α) (a : α) :

The embedding map is always a semi-contraction.

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

Every separable metric space embeds isometrically in ℓ_infty_ℝ.

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

Equations

The Kuratowski embedding is an isometry.