TheoremStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedaudited 2026-09-22

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MA plus not-CH yields a normal nonmetrizable Moore space

Statement

$Z F C + M A + \neg C H$ proves that there is a separable normal nonmetrizable Moore space.

Facts & Assumptions

Given: $M A + \neg C H$ and a subset $E_{0} \subseteq R$ with $∣ E_{0} ∣ = ω_{1}$ .

[F1]

Every set of reals of cardinality $ω_{1}$ is a Q-set and an uncountable Q-set exists (Martin's axiom produces an uncountable Q-set, Q-sets and Bing's tangent-disk Moore-space interface).

[F2]

For every uncountable Q-set $E$ the tangent-disk space $Z (E)$ is a separable normal nonmetrizable Moore space (Bing's Q-set space is a normal nonmetrizable Moore space, Q-sets and Bing's tangent-disk Moore-space interface, Moore spaces and developments).

Proof

technique · direct

1.1

By [F1] the set $E_{0}$ , of cardinality $ω_{1}$ , is an uncountable Q-set.

givenF1

2.1

Applying [F2] to $E_{0}$ produces a separable normal nonmetrizable Moore space, namely the tangent-disk space $Z (E_{0})$ .

step 1.1F2∎

Remarks

This is the second of the two standard refutations of the normal Moore space conjecture, and the only one available at $ω_{1}$ . It uses no large cardinal, only $M A$ and the failure of $C H$ ; the construction is separable, in contrast with the $C H$ construction recorded elsewhere on this page.
The two ingredients are independent. The Q-set comes from Martin's axiom through almost-disjoint forcing (Martin's axiom extends families almost disjoint from a subfamily), and the space comes from Bing's tangent-disk construction; the theorem spends both.

Depends on

Used by

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Sources

Dennis K. Burke, The Normal Moore Space Problem (standard reference, not scraped)