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TheoremStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedaudited 2026-09-22
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PMEA implies the normal Moore space conjecture

Statement

ZFC+PMEA proves the normal Moore space conjecture: every normal Moore space is metrizable. Already PMEA-σ suffices (PMEA and PMEA-sigma, Moore spaces and developments, Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not).

Facts & Assumptions

Given: A normal Moore space X and PMEA-σ. (The formally stronger PMEA hypothesis in the first sentence of the Statement supplies PMEA-σ by [F4].)

[F1]

Every Moore space is first countable: a development supplies at each point the countable star family as a local base (Moore spaces and developments).

[F2]

Under PMEA-σ, every first countable normal space is collectionwise normal (PMEA makes normal low-character spaces collectionwise normal).

[F3]

Every collectionwise normal Moore space is metrizable (Collectionwise normal Moore spaces are metrizable).

[F4]

PMEA implies PMEA-σ, since a c-additive full extension is countably additive (PMEA and PMEA-sigma).

Proof

technique · direct
1.1givenF1

The space X is first countable by [F1] and is normal and Moore by hypothesis.

2.1step 1.1F2

By [F2] the space X is collectionwise normal.

3.1step 2.1F3F4∎

By [F3] the collectionwise normal Moore space X is metrizable; since PMEA implies PMEA-σ by [F4], the argument used only PMEA-σ.

Remarks

  • This is Nyikos' provisional solution in Fremlin's form. The measure-theoretic input is PMEA-σ alone; the topological input is that a first countable normal space is collectionwise normal under it; the metrization of collectionwise normal Moore spaces is the separate local theorem of this page.

Depends on

Used by

Dependency tree · two levels

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Sources