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 has a $\sigma$-discrete basis

Proof

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Source of Name

This entry was named for R.H. Bing.

Historical Note

Bing's Metrization Theorem was discovered in $1951$ by R.H. Bing, 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