# Documentation

Mathlib.Topology.MetricSpace.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.

• exists_pseudo_metric : m, UniformSpace.toTopologicalSpace = t

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 : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X.

Instances

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

Instances For
instance TopologicalSpace.pseudoMetrizableSpace_prod {X : Type u_2} {Y : Type u_3} [] [] :
theorem Inducing.pseudoMetrizableSpace {X : Type u_2} {Y : Type u_3} [] [] {f : XY} (hf : ) :

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

Every pseudo-metrizable space is first countable.

instance TopologicalSpace.pseudoMetrizableSpace_pi {ι : Type u_1} {π : ιType u_4} [] [(i : ι) → TopologicalSpace (π i)] [∀ (i : ι), ] :
• exists_metric : m, UniformSpace.toTopologicalSpace = t

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 : MetricSpace X := TopologicalSpace.metrizableSpaceMetric X.

Instances
instance MetricSpace.toMetrizableSpace {X : Type u_5} [m : ] :
noncomputable def TopologicalSpace.metrizableSpaceMetric (X : Type u_5) [] :

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

Instances For
instance TopologicalSpace.metrizableSpace_prod {X : Type u_2} {Y : Type u_3} [] [] :
theorem Embedding.metrizableSpace {X : Type u_2} {Y : Type u_3} [] [] {f : XY} (hf : ) :

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

instance TopologicalSpace.metrizableSpace_pi {ι : Type u_1} {π : ιType u_4} [] [(i : ι) → TopologicalSpace (π i)] [∀ (i : ι), ] :
theorem TopologicalSpace.exists_embedding_l_infty (X : Type u_2) [] [] :
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.