The normal Moore space conjecture is independent, and its consistency needs a large cardinal

A Moore space is a regular space carrying a development: a sequence $(\mathcal{G}_n)_{n \in \mathbb{N}}$ of open covers such that for each point $x$ the collection of stars $\mathrm{St}(x, \mathcal{G}_n)$ is a neighbourhood base at $x$. The normal Moore space conjecture (NMSC) asserts that every normal Moore space is metrisable.

NMSC is not decided by ZFC, and its two sides have very different costs.

(a) It fails under CH. Fleissner (1982) constructs a normal nonmetrisable Moore space from the continuum hypothesis. Since CH holds in the constructible universe, NMSC fails in $L$. Non-metrisable normal Moore spaces also exist under MA + (not CH), so both of the standard opposing hypotheses of Martin's Axiom ‡ refute it.

(b) It holds under PMEA. Nyikos (1980) proves that the product measure extension axiom, that the usual product measure on $\{0,1\}^{\kappa}$ extends to a measure on all subsets, implies every normal Moore space is metrisable. PMEA is consistent relative to the existence of a strongly compact cardinal.

(c) A large cardinal is necessary, not just convenient. Fleissner (1982) proves that if every normal Moore space is metrisable then there is an inner model with a measurable cardinal. So the consistency of NMSC is not provable from the consistency of ZFC alone.

• Not proved in this library. None of (a), (b), or (c) is proved here. The library now has a metrisation track and the standard metrisation theorems, but not the large-cardinal, forcing, measure-theoretic or inner-model machinery needed for these independence statements.

• What would prove it. For (a), a CH construction plus the standard metrisation theorems. For (b), large-cardinal theory (strongly compact cardinals) and the forcing that produces PMEA, plus measure theory. For (c), inner-model theory and the covering lemma. Two deferred tracks meet here, set theory beyond choice and measure theory.

• Why it matters here. It is the cleanest example in general topology of a natural question whose answer is not merely independent but genuinely expensive: unlike CH (The continuum hypothesis, and what this page does not prove ↗), one side of it cannot be obtained from Con(ZFC) at all. Any metrisation page in this library must therefore state Bing's and Nagata-Smirnov's theorems and stop; the tempting further step, dropping collectionwise normality to plain normality, is not available and cannot be made available by working harder.

• Conditional discipline. (a) and (b) are implications from stated hypotheses; the consistency of PMEA is relative to a large-cardinal hypothesis strictly stronger than Con(ZFC), and (c) is an implication between consistency strengths. Nothing here asserts that measurable or strongly compact cardinals exist.