by Franklin Tall
Set-theoretic topology was inspired by Mary Ellen Rudin for three decades. Her influence and leadership in North America and worldwide were central to the development of the field.
Mary Ellen grew up in a small town in Texas with no idea that she might become a mathematician. When R. L. Moore was trawling for prospects at registration at the University of Texas, he sensed she was a good one and began reeling her in. In classic Moore fashion, he created a confident, powerful researcher who didn’t know any mathematics other than what he led her to discover. Her thesis and first few papers are written in Moore’s antiquated language, but Steve Watson [e4] has demonstrated that they are still well worth reading. In her thesis [1], she gave the first example of a nonseparable Moore space satisfying the countable chain condition. After positions at Duke (where she met her husband, Walter Rudin) and Rochester, the Rudins moved to the University of Wisconsin at Madison. The UW system had a nepotism regulation that prevented Mary Ellen from having a regular position because Walter had one. As a result, she stayed a “lecturer” which had one advantage: she was free of committee work. In 1971 the regulation was finally abolished, and she was promoted directly to full professor. I first met Mary Ellen in the summer of 1967. I was casting around for a supervisor and Mary Ellen was very approachable, helpful, and encouraging — qualities that supported a long line of students thereafter. I was her first PhD student, and in 1969 I had to have the department chair cosign my thesis because, as a lecturer, Mary Ellen was not allowed to be a full-fledged supervisor.
The first paper of Mary Ellen that most set-theoretic topologists have heard of is [2], in which she used a Suslin tree to construct a Dowker space. Such a construction would be quite interesting if done for the first time now, but in 1955, before the consistency of the existence of such trees was known, this was quite extraordinary. She constructed a “real” Dowker space in [3], leading to an invitation to address the ICM.
The students of Moore such as Mary Ellen and R. H. Bing were familiar with both elementary set-theoretic techniques and geometric ones, especially dealing with connected spaces. Mary Ellen’s geometric abilities proved invaluable in her work on pathological manifolds. Her most noteworthy result in this area is the consistency (with P. Zenor ) and independence of the existence of perfectly normal nonmetrizable manifolds [5], [7]. Some other areas wherein she produced significant results are: classifying ultrafilters on the natural numbers (the Rudin–Keisler and Rudin–Frolik orders), showing (with Y. Benyamini and her student Mike Wage) that continuous images of Eberlein compacts (weakly compact subspaces of Banach spaces) are also Eberlein [6], and establishing many theorems about normality of products and box products. Let me give just one example of a normality theorem, consistently solving a problem of K. Morita. Mary Ellen, with her student Amer Bešlagić, proved in [8]: Gödel’s Axiom of Constructibility implies that for a space \( X \) whose product with every metrizable space is normal, \( X \) is metrizable if and only if \( X \times Z \) is normal, for every space \( Z \) such that \( Z \)’s products with metrizable spaces are all normal. Amazingly, they proved this by constructing an example!
Mary Ellen’s fame was largely as a producer of weird and wonderful topological spaces — examples and counterexamples. She had an uncanny ability to start off with a space that had some of the properties she wanted and then push it and pull it until she got exactly what she wanted. In her heyday, several times a week she would receive inquiries from mathematicians around the world — often nontopologists — asking for an example of something or other. She would answer these by writing back on the back of the same letter, not from “green” sentiments, but to avoid clutter. I am convinced that her avoidance of clutter, especially mentally, was one of the secrets of her mathematical power. By not filling up her memory, she maximized processing capability in her brain.
Mary Ellen’s papers became (and stayed) more set-theoretic, starting in the late 1960s. There were a variety of reasons for this in addition to what Marxists might call “historical necessity.” The most important reason was that set theory was being done at Wisconsin, most notably by Ken Kunen, who arrived in 1968. David Booth and I were both students in logic who had begun thinking about applications of set theory to topology; we frequently went back and forth between Ken and Mary Ellen, and that accelerated the collaboration that made Madison such an exciting place for set-theoretic topology in the 1970s. During the summers I and many other set-theoretic topologists would come to visit and interact with Mary Ellen and Ken. The most memorable summer, though, was the one of 1974, when we all went off to the CBMS Regional Conference in Laramie, Wyoming, for Mary Ellen’s lectures on set-theoretic topology [4]. I remember the sense of excitement we felt at Mary Ellen’s lectures, which set the course of the field for the next decade. The resulting book, with chapters on cardinal functions, hereditary separability versus hereditary Lindelöfness, box products, and so forth, established a framework for the new field of set-theoretic topology, which grew out of general topology but employed new methods, which led to new questions. It was an exciting time. It was a noteworthy challenge to try to solve a problem on her problem list.
We celebrated Mary Ellen in 1991 on the occasion of her retirement with a conference in Madison. The proceedings were published in [e2]; much of this note is drawn from my contribution [e3] in [e2], where more details of her mathematics can be found. There is also the excellent biographical article [e1]. Mary Ellen’s research certainly did not stop with her retirement. Particularly impressive was her complex proof of Nikiel’s Conjecture, characterizing the continuous images of compact ordered spaces as the compact monotonically normal spaces [9]. As Steve Watson wrote in [e4] (quoted by Todd Eisworth in his review [e5] of [9]), “Reading the articles of Mary Ellen Rudin, studying them until there is no mystery takes hours and hours; but those hours are rewarded, the student obtains power to which few have access. They are not hard to read, they are just hard mathematics, that’s all.”
Although her body failed her, her mind remained sharp as she aged. She stopped traveling, and I was busy with family responsibilities, so I saw little of her in recent years, but she remained and remains an inspiration for all of us in the field.
