TheoremStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedaudited 2026-09-22‡ rests on unproved material (inherited)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 2 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Under choice, every metric space has a σ-discrete basis and ‡ Under choice, a regular T₁ space with a σ-locally-finite basis has a compatible normal sequence. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

PMEA implies the normal Moore space conjecture

Statement

$Z F C + P M E A$ 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

1.1

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

givenF1

2.1

By [F2] the space $X$ is collectionwise normal.

step 1.1F2

3.1

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

step 2.1F3F4∎

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

42 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

D. H. Fremlin, Real-valued-measurable cardinals (standard reference, not scraped)
Dennis K. Burke, The Normal Moore Space Problem (standard reference, not scraped)