Metrization of Regular Second Countable Space

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Theorem

Let $T = \struct {S, \tau}$ be a $T_1$ space.


The following statements are equivalent:

$(1): \quad T$ is regular and second-countable
$(2): \quad T$ is homeomorphic to a metric subspace of the Hilbert cube $I^\omega$
$(3): \quad T$ is metrizable and separable


Proof

Condition $(1)$ implies Condition $(2)$

Follows immediately from Regular Second-Countable Space is Homeomorphic to Subspace of Hilbert Cube.

$\Box$

Condition $(2)$ implies Condition $(3)$

Let $T$ be homeomorphic to a subspace of the Hilbert cube $\struct{I^\omega, d_2}$ where $d_2$ is the metric: $\ds \forall x = \sequence {x_i}, y = \sequence {y_i} \in I^\omega: \map {d_2} {x, y} := \paren {\sum_{k \mathop \in \N_{>0} } \paren {x_k - y_k}^2}^{\frac 1 2}$


By definition of metrizable topology:

$\struct{I^\omega, \tau_{d_2}}$ is metrizable

where $\tau_{d_2}$ is the topology induced by $d_2$.


From Subspace of Metrizable Space is Metrizable Space:

$T$ is homeomorphic to a metrizable space

From Topological Space Homeomorphic to Metrizable Space is Metrizable Space:

$T$ is metrizable


From Hilbert Cube is Separable:

the Hilbert cube $\struct{I^\omega, d_2}$ is separable

From Subspace of Separable Metric Space is Separable:

$T$ is homeomorphic to a separable space

From Continuous Image of Separable Space is Separable:

$T$ is separable

$\Box$

Condition $(3)$ implies Condition $(1)$

Let $T$ be metrizable and separable.


By definition of metrizable space:

there exists a metric $d$ on $S$ such that the topology $\tau$ on $S$ is the topology induced by $d$

From Metric Space is Separable iff Second-Countable

$\struct{S, d}$ is a second-countable space


From Metric Space is Fully Normal:

$\struct{S, \tau}$ is a fully normal space

From Fully Normal Space is Normal Space:

$\struct{S, \tau}$ is a normal space

From Normal Space is Regular Space:

$\struct{S, \tau}$ is a regular space

$\blacksquare$


Also known as

In some sources this theorem is known as Urysohn's Metrization Theorem as it includes the sufficient condition to the necessary conditions of metrization that is the subject of what is more commonly known as Urysohn's Metrization Theorem.


Also see


Sources

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