# mathlibdocumentation

topology.metric_space.metrizable

# Metrizability of a T₃ topological space with second countable topology #

In this file we define metrizable topological spaces, i.e., topological spaces for which there exists a metric space structure that generates the same topology.

We also show that a T₃ topological space with second countable topology X is metrizable.

First we prove that X can be embedded into l^∞, then use this embedding to pull back the metric space structure.

@[class]
structure topological_space.pseudo_metrizable_space (X : Type u_5) [t : topological_space X] :
Prop
• exists_pseudo_metric :

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 : pseudo_metric_space X := topological_space.pseudo_metrizable_space_pseudo_metric X.

Instances of this typeclass
@[protected, instance]
noncomputable def topological_space.pseudo_metrizable_space_pseudo_metric (X : Type u_1)  :

Construct on a metrizable space a metric compatible with the topology.

Equations
@[protected, instance]
def topological_space.pseudo_metrizable_space_prod {X : Type u_2} {Y : Type u_3}  :
theorem inducing.pseudo_metrizable_space {X : Type u_2} {Y : Type u_3} {f : X → Y} (hf : inducing f) :

Given an inducing map of a topological space into a pseudo metrizable space, the source space is also pseudo metrizable.

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Every pseudo-metrizable space is first countable.

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@[protected, instance]
def topological_space.pseudo_metrizable_space_pi {ι : Type u_1} {π : ι → Type u_4} [finite ι] [Π (i : ι), topological_space (π i)] [∀ (i : ι), ] :
@[class]
structure topological_space.metrizable_space (X : Type u_5) [t : topological_space X] :
Prop
• exists_metric : ∃ (m : ,

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 : metric_space X := topological_space.metrizable_space_metric X

Instances of this typeclass
@[protected, instance]
def metric_space.to_metrizable_space {X : Type u_1} [m : metric_space X] :
@[protected, instance]
noncomputable def topological_space.metrizable_space_metric (X : Type u_1)  :

Construct on a metrizable space a metric compatible with the topology.

Equations
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@[protected, instance]
def topological_space.metrizable_space_prod {X : Type u_2} {Y : Type u_3}  :
theorem embedding.metrizable_space {X : Type u_2} {Y : Type u_3} {f : X → Y} (hf : embedding f) :

Given an embedding of a topological space into a metrizable space, the source space is also metrizable.

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def topological_space.metrizable_space.subtype {X : Type u_2} (s : set X) :
@[protected, instance]
def topological_space.metrizable_space_pi {ι : Type u_1} {π : ι → Type u_4} [finite ι] [Π (i : ι), topological_space (π i)] [∀ (i : ι), ] :
theorem topological_space.exists_embedding_l_infty (X : Type u_2) [t3_space X]  :
∃ (f : ,

A T₃ topological space with second countable topology can be embedded into l^∞ = ℕ →ᵇ ℝ.

Urysohn's metrization theorem (Tychonoff's version): a T₃ topological space with second countable topology X is metrizable, i.e., there exists a metric space structure that generates the same topology.

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