# Bing's Metrization Theorem

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## 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

## Proof

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## 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**.

## Sources

- 1955: John L. Kelley:
*General Topology*: Chapter $4$: Embedding and Metrization - 1975: James R. Munkres:
*Topology*: Chapter $6$: Metrization Theorems and Paracompactness: $\S40$: The Nagata-Smirnov Metrization Theorem: Exrcise: $5$ - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.): Part $\text {III}$: Metrization Theory: Conjectures and Counterexamples: Screenable Spaces