# Bing's Metrization Theorem

## Theorem

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

Then:

$T$ is metrizable if and only if $T$ is regular and $T_0$ and has a $\sigma$-discrete basis

## Source of Name

This entry was named for R.H. Bing.

## Historical Note

The theorem by R.H. Bing was discovered in $1951$ independently of the Nagata-Smirnov Metrization Theorem discovered by Jun-iti Nagata ($1950$) and Yurii Mikhailovich Smirnov ($1951$).

The two theorems Bing's Metrization Theorem and Nagata-Smirnov Metrization Theorem are often merged as the Bing-Nagata-Smirnov Metrization Theorem.