Math 2.0

General topology Metrization Problem

Normal Moore space problem

Builds on Moore spaceNormal spaceMetrizable space

Problem

Is every normal Moore space metrizable?

Remarks

Every metrizable space is a normal Moore space, so the question asks whether the converse holds. Normality is the key condition: the Moore plane is a Moore space that is not normal and not metrizable. Under Martin's axiom together with the negation of the continuum hypothesis there is a separable normal Moore space that is not metrizable. So a positive answer cannot be proved in ZFC, the usual axioms of set theory. Peter Nyikos showed that the product measure extension axiom implies that every normal Moore space is metrizable. That axiom is consistent if a strongly compact cardinal is consistent. William Fleissner showed that the positive answer implies the consistency of a measurable cardinal. So a large cardinal is needed for the positive answer, and the problem is settled only relative to such axioms.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph
Metrizable space, concept, no evidence levelMetrizable spaceconceptMoore space, concept, no evidence levelMoore spaceconceptNormal space, concept, no evidence levelNormal spaceconceptNormal Moore space problem, problem, no evidence levelNormal Moore space pr…problem
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. Metrizable space concept
  2. Moore space concept
  3. Normal space concept
  4. Normal Moore space problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Moore_space_(topology)
About this page
TagZ6T00E3JQH
URIm2://Z6T00E3JQH
Kindproblem
WritingW0 stub
MSC54E30, 54A35, 03E35
StatusThis copy of the site has no build data. Run m2 status Z6T00E3JQH.
AboutMoore space, Metrizable space
UsesMoore space, Normal space, Metrizable space
Changed2 November 2026

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