# Category:Metrization Theorems

This category contains pages concerning Metrization Theorems:

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

## Subcategories

This category has only the following subcategory.

## Pages in category "Metrization Theorems"

The following 7 pages are in this category, out of 7 total.