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**Mary Ellen Rudin’s contributions to the theory of nonmetrizable manifolds.**
*(English)*
Zbl 0894.54003

Tall, Franklin D. (ed.), The work of Mary Ellen Rudin. Papers presented at the seventh summer conference on general topology and applications in honor of Mary Ellen Rudin and her work, held in Madison, WI, USA, from June 26 to 29, 1991. New York, NY: The New York Academy of Sciences. Ann. N. Y. Acad. Sci. 705, 92-113 (1993).

There are still any number of books and articles that define “manifold” to mean “metrizable space, in which every point has a neighborhood homeomorphic to \(\mathbb{R}^n\) for some finite \(n\),” or even require that the space be compact. But general topologists for at least the last fifteen years have mentally substituted “Hausdorff” for “metrizable”. This change was wrought by a single unexpected result: in 1975, Mary Ellen Rudin provided consistent solutions to a problem posed by R. L. Wilder [Topology of manifolds (1949; Zbl 0039.39602)]. The problem was whether every perfectly normal generalized manifold is metrizable, and Mary Ellen showed that under the axiom \(\diamond\) (a consequence of \(V=L\), Gödel’s Axiom of Constructibility) there is a perfectly normal, hereditarily separable, countably compact, nonmetrizable 2-manifold. By sacrificing countable compactness, she built an example using the continuum hypothesis (CH) alone, which P. L. Zenor helped to simplify [M. E. Rudin and P. Zenor, Houston J. Math. 2, 129-134 (1976; Zbl 0315.54028)]. Then, in 1978, Mary Ellen showed that the existence of a perfectly normal nonmetrizable manifold is independent of the usual axioms of set theory. Specifically, she showed \(\text{MA}(\omega_1)\) implies that all perfectly normal manifolds are metrizable. This did not, as has generally been believed until now, solve completely the Wilder problem as originally stated, although Wilder was quite satisfied with the solution and perhaps no longer cared about his original problem. Nor did either discovery really touch a 1935 problem of Alexandroff. This is not to take away from the magnitude of Mary Ellen’s discovery. In fact, the first result takes on added luster for answering Wilder’s question (at least consistently) in an especially strong way: with an example that is a manifold and not just a “generalized” manifold. More importantly, her discoveries have inspired a spate of research in the topic of nonmetrizable manifolds by several topologists, myself included. In fact, the modern study of nonmetrizable manifolds can be said to begin with her results, in the same basic sense that the modern study of set-theoretic topology can be said to begin with the 1967 realization that \(\text{MA}(\omega_1)\) implies the existence of separable nonmetrizable normal Moore spaces. Elaborating a bit on the last theme, there are still large areas of the theory of nonmetrizable generalized manifolds that are “stuck in the classical period.” This represents a challenge to future research that I will make more explicit in the final section, which gives not just open problems but also whole research topics.

In between, I will be mostly expounding on Mary Ellen Rudin’s research into this area and the results leading to them. I will also make some effort to bring the reader up to date on the research into nonmetrizable manifolds that has been done since the Handbook article on manifolds [P. Nyikos, The theory of nonmetrizable manifolds, in ‘Handbook of set-theoretic topology’, 633-684 (1984; Zbl 0583.54002)], although space does not permit much more than a summary of results.

For the entire collection see [Zbl 0801.00017].

In between, I will be mostly expounding on Mary Ellen Rudin’s research into this area and the results leading to them. I will also make some effort to bring the reader up to date on the research into nonmetrizable manifolds that has been done since the Handbook article on manifolds [P. Nyikos, The theory of nonmetrizable manifolds, in ‘Handbook of set-theoretic topology’, 633-684 (1984; Zbl 0583.54002)], although space does not permit much more than a summary of results.

For the entire collection see [Zbl 0801.00017].

### MSC:

54-03 | History of general topology |

54A35 | Consistency and independence results in general topology |

57P99 | Generalized manifolds |

54E30 | Moore spaces |

54D15 | Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.) |

01A70 | Biographies, obituaries, personalia, bibliographies |

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\textit{P. J. Nyikos}, in: The work of Mary Ellen Rudin. Papers presented at the seventh summer conference on general topology and applications in honor of Mary Ellen Rudin and her work, held in Madison, WI, USA, from June 26 to 29, 1991. New York, NY: The New York Academy of Sciences. 92--113 (1993; Zbl 0894.54003)