---
tag: Z6T00E3JQH
kind: problem
name: Normal Moore space problem
msc: ["54E30", 54A35, "03E35"]
uses: [68MA0RM5P6, 0NSR9V8TED, 091GJ3FYD7]
about: [68MA0RM5P6, 091GJ3FYD7]
sources:
  - url: https://en.wikipedia.org/wiki/Moore_space_(topology)
history:
  posed_by: "F. Burton Jones"
  first_stated: "1937"
---

## Problem

Is every [[normal|0NSR9V8TED]] [[Moore space|68MA0RM5P6]] [[metrizable|091GJ3FYD7]]?

## 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|SAR7PBYF07]] 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.

## Explanations

### L1 human

Moore spaces look very much like metric spaces, and normality is one of the main properties of metric spaces.
The question asks whether these two together are enough to make the space metric.
Work on it showed that the answer depends on extra axioms of set theory, which made it a model case for the use of set theory in topology.