Math 2.0

General topology Metrization Theorem

Bing metrization theorem

Builds on Topological spaceRegular spaceHausdorff spaceMetrizable space

Statement

Let XX be a topological space. A family ℱ\mathcal{F} of subsets of XX is discrete if each point of XX has a neighbourhood that meets at most one member of ℱ\mathcal{F}. A family is σ\sigma-discrete if it is a union of countably many discrete families. Then XX is metrizable if and only if XX is regular, Hausdorff and has a σ\sigma-discrete base.

Remarks

A discrete family is locally finite, so one direction follows from the Nagata–Smirnov metrization theorem. For the other direction, take a metric space. The cover by balls of radius 1/n1/n has a σ\sigma-discrete open refinement, built with a well-order of the balls. The union over nn of these refinements is a σ\sigma-discrete base. In the real line, the open intervals with rational endpoints form a countable base, and a countable base is σ\sigma-discrete.

Going deeper

For a student

The theorems of Bing and of Nagata–Smirnov were found independently and give the same class of spaces. Bing's form is often handy in proofs, because a discrete family behaves like a disjoint family of sets that are also kept apart from each other. In the same work Bing proved that a collectionwise normal Moore space is metrizable.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Hausdorff space, concept, no evidence levelHausdorff spaceconceptMetrizable space, concept, no evidence levelMetrizable spaceconceptRegular space, concept, no evidence levelRegular spaceconceptTopological space, concept, no evidence levelTopological spaceconceptBing metrization theorem, statement, no evidence levelBing metrization theo…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Hausdorff space concept
  2. Metrizable space concept
  3. Regular space concept
  4. Topological space concept
  5. Bing metrization theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Bing_metrization_theorem
About this page
TagRGCZ2P9QRM
URIm2://RGCZ2P9QRM
Kindstatement, theorem
WritingW0 stub
MSC54E35
StatusThis copy of the site has no build data. Run m2 status RGCZ2P9QRM.
AboutMetrizable space
UsesTopological space, Regular space, Hausdorff space, Metrizable space
Same as
Changed2 November 2026

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