CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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V=L refutes the normal Moore space conjecture

Statement

$Z F C + V = L$ refutes the normal Moore space conjecture: it proves that there is a normal nonmetrizable Moore space (CH yields a normal nonmetrizable Moore space, Moore spaces and developments).

Facts & Assumptions

Given: The axiom $V = L$ over $Z F C$ .

[F1]

$V = L$ implies $A C + G C H$ (The generalized continuum hypothesis holds in L).

[F2]

GCH gives the continuum hypothesis at $ω$ : the instance at the infinite cardinal $ℵ_{0}$ says $2^{ℵ_{0}} = ℵ_{1}$ , since $ℵ_{0}^{+} = ℵ_{1}$ and $P (ω)$ has cardinality $2^{ℵ_{0}}$ (Cantor's theorem: $A ≺ P (A)$ , Fleissner's HYP covering interface); this is the form of CH consumed by CH yields a normal nonmetrizable Moore space.

[F3]

$Z F C + C H$ proves that a normal nonmetrizable Moore space exists (CH yields a normal nonmetrizable Moore space).

Proof

technique · direct

1.1givenF1

Assume $V = L$ . By [F1] both AC and GCH hold.

2.1step 1.1F2

By [F2], GCH gives $2^{ℵ_{0}} = ℵ_{1}$ , i.e. CH.

3.1step 2.1F3

By [F3], applied under CH, there is a normal nonmetrizable Moore space.

4.1step 3.1given∎

A normal Moore space that is not metrizable is a counterexample to the normal Moore space conjecture, so $V = L$ refutes it.

Depends on

Used by

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Dependency tree · two levels

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Sources

Fleissner, Normal nonmetrizable Moore space from continuum hypothesis or nonexistence of inner models with measurable cardinals (standard reference, not scraped)