The normal Moore space conjecture and large cardinals.

*(English)*Zbl 0562.54039
Handbook of set-theoretic topology, 733-760 (1984).

[For the entire collection see Zbl 0546.00022.]

In 1937, F. Burton Jones proved that the axiom \(2^{\aleph_ 0}<2^{\aleph_ 1}\) (now known to be independent of the ZFC-axioms) implies that every separable normal Moore-space is metrizable. Ever since then, the ”normal Moore space problem”, of whether every normal Moore space is metrizable, has been one of the most famous and stimulating problems in set-theoretic topology. Many prominent topologists contributed fascinating results to the vast literature on this problem (Bing, Alexandroff, Arkhangel’skij, Heath, Nyikos, Tall, Fleischner, to name only a few). Surveys (including highlights of the problems history) can be found in Peter Nyikos’ paper ”A provisional solution to the normal Moore space problem” [Proc. Am. Math. Soc. 78, 429-435 (1980; Zbl 0446.54030)], further, in two articles by M. E. Rudin and a paper of F. Tall (both cited explicitly in Nyikos’ paper) and in F. Tall’s article in this handbook. The normal Moore space problem is strongly connected with basic set-theoretic tools. By an example of the author’s, yielding a non- metrizable normal Moore space under \(MA+\neg CH\), we know that the existence of such a space is consistent with ZFC. Since - on the other hand - P. Nyikos has shown that the product measure extension axiom (PMEA) implies that every normal Moore space is metrizable, and knowing of Kunen’s result that PMEA is consistent relative to a strongly compact cindimal, we know that the normal Moore space problem might be strongly connected with large cardinals. And in fact, in 1983 Fleissner proved that ”if all normal Moore spaces are metrizable, then there is an inner model with a measurable cardinal” [Trans. Am. Math. Soc. 273, 365-373 (1983; Zbl 0498.54025)]. So it might become clear why ”large cardinals” are included in the title of the present article, which provides an excellent survey of the recent history and backgrounds of the normal Moore space problem, of PMEA and all the set- and model-theoretic tools and results around it. It is worthwhile mentioning that the author not only collects results but in most cases (e.g. Nyikos’ theorem or Kunen’s result that Con (\(\exists\) strongly compact cardinal)\(\to Con (PMEA)\), and others) also explains the idea of the proof or includes and partially improves the parts and the ideas connected with them. The author presents explicit descriptions and modifications of the famous normal nonmetrizable Moore spaces and normal not collectionwise normal spaces, including the space ”George”, ”Navy’s space” and others.

In summary, we can say that the present article contains an excellent survey and - having also the character of a research paper - it moreover provides a clear and well-written introduction to this field.

In 1937, F. Burton Jones proved that the axiom \(2^{\aleph_ 0}<2^{\aleph_ 1}\) (now known to be independent of the ZFC-axioms) implies that every separable normal Moore-space is metrizable. Ever since then, the ”normal Moore space problem”, of whether every normal Moore space is metrizable, has been one of the most famous and stimulating problems in set-theoretic topology. Many prominent topologists contributed fascinating results to the vast literature on this problem (Bing, Alexandroff, Arkhangel’skij, Heath, Nyikos, Tall, Fleischner, to name only a few). Surveys (including highlights of the problems history) can be found in Peter Nyikos’ paper ”A provisional solution to the normal Moore space problem” [Proc. Am. Math. Soc. 78, 429-435 (1980; Zbl 0446.54030)], further, in two articles by M. E. Rudin and a paper of F. Tall (both cited explicitly in Nyikos’ paper) and in F. Tall’s article in this handbook. The normal Moore space problem is strongly connected with basic set-theoretic tools. By an example of the author’s, yielding a non- metrizable normal Moore space under \(MA+\neg CH\), we know that the existence of such a space is consistent with ZFC. Since - on the other hand - P. Nyikos has shown that the product measure extension axiom (PMEA) implies that every normal Moore space is metrizable, and knowing of Kunen’s result that PMEA is consistent relative to a strongly compact cindimal, we know that the normal Moore space problem might be strongly connected with large cardinals. And in fact, in 1983 Fleissner proved that ”if all normal Moore spaces are metrizable, then there is an inner model with a measurable cardinal” [Trans. Am. Math. Soc. 273, 365-373 (1983; Zbl 0498.54025)]. So it might become clear why ”large cardinals” are included in the title of the present article, which provides an excellent survey of the recent history and backgrounds of the normal Moore space problem, of PMEA and all the set- and model-theoretic tools and results around it. It is worthwhile mentioning that the author not only collects results but in most cases (e.g. Nyikos’ theorem or Kunen’s result that Con (\(\exists\) strongly compact cardinal)\(\to Con (PMEA)\), and others) also explains the idea of the proof or includes and partially improves the parts and the ideas connected with them. The author presents explicit descriptions and modifications of the famous normal nonmetrizable Moore spaces and normal not collectionwise normal spaces, including the space ”George”, ”Navy’s space” and others.

In summary, we can say that the present article contains an excellent survey and - having also the character of a research paper - it moreover provides a clear and well-written introduction to this field.

Reviewer: H.-C.Reichel

##### MSC:

54E30 | Moore spaces |

54E35 | Metric spaces, metrizability |

03E55 | Large cardinals |

54-02 | Research exposition (monographs, survey articles) pertaining to general topology |

03E35 | Consistency and independence results |

54A35 | Consistency and independence results in general topology |