Moore Space (topology) - Normal Moore Space Conjecture

Normal Moore Space Conjecture

For a long time, topologists were trying to prove the so-called normal Moore space conjecture: every normal Moore space is metrizable. This was inspired by the fact that all known Moore spaces that were not metrizable were also not normal. This would have been a nice metrization theorem. There were some nice partial results at first; namely properties 7, 8 and 9 as given in the previous section.

Here we see that we drop metacompactness from Traylor's theorem, but at the cost of a set-theoretic assumption. Another example of this is Fleissner's theorem that the axiom of constructibility implies that locally compact, normal Moore spaces are metrizable.

On the other hand, under the Continuum hypothesis (CH) and also under Martin's Axiom and not CH, there are several examples of non-metrizable normal Moore spaces. Nyikos proved that, under the so-called PMEA (Product Measure Extension Axiom), which needs a large cardinal, all normal Moore spaces are metrizable. Finally, it was shown later that any model of ZFC in which the conjecture holds, implies the existence of a model with a large cardinal. So large cardinals are needed essentially.

Moore himself proved the theorem that a collectionwise normal Moore space is metrizable, so strengthening normality is another way to settle the matter.

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