Metrization Theorems

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Theorem

Nagata-Smirnov Metrization Theorem

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


Then:

$T$ is metrizable if and only if $T$ is regular and has a basis that is $\sigma$-locally finite.


Bing's Metrization Theorem

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


Then:

$T$ is metrizable if and only if $T$ is regular and has a $\sigma$-discrete basis


Smirnov Metrization Theorem

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


Then:

$T$ is metrizable if and only if $T$ is paracompact and locally metrizable.


Urysohn's Metrization Theorem

Let $T = \struct {S, \tau}$ be a topological space which is regular and second-countable.


Then $T$ is metrizable.


Frink's Metrization Theorem

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


Then:

$T$ is metrizable

if and only if:

for all $s \in S$ there exists a countable neighborhood basis, denoted $\set {U_{s, n} : n \in \N}$, such that:
$(1): \quad \forall s \in S, n \in \N : U_{s, n + 1} \subseteq U_{s, n}$
$(2): \quad \forall s \in S, n \in \N : \exists m > n : \forall t \in S : U_{t, m} \cap U_{s ,m} \ne \O \implies U_{t, m} \subseteq U_{s, n}$