**Mary Ellen Rudin**'s name before she married was Mary Ellen Estill. Her parents were Joe Jefferson Estill, who was a civil engineer, and Irene Shook, who was an English teacher in a High School before marrying Joe. For ten years Mary Ellen was an only child until her younger brother was born. Both of Mary Ellen's grandmothers had graduated from Mary Sharp College in Winchester, Tennessee. The tradition of education was strongly felt in the family but they had no special espectations for Mary Ellen and certainly none in mathematics.

Joe Estill was engaged on road building projects around Leakey, Texas and it was in that small isolated town that Mary Ellen was brought up. Certainly an engineer to improve the roads was vital. To reach Leakey in the 1920s was certainly not easy with the only route being a 50 mile dirt road through a canyon which forded the Frio River seven times. Uvalde is situated at the bottom of the canyon and the road travelled north from there up the Frio river, then beyond Leakey to the town of Junction which is as far north of Leakey as Uvalde is to the south. Mary Ellen spoke of her childhood in Leakey:-

We had few toys. There was no movie house in town. We listened to the radio. But our games were very elaborate and purely in the imagination. I think actually that that is something that contributes to making a mathematician - having time to think and being in the habit of imagining all sorts of complicated things.

Mary Ellen attended the Leakey school, the only school of any kind within 60 miles of Leakey. It had 10 grades altogether with 2 ^{1}/_{2} teachers for its "high school" where each grade had about 5 students. When she entered the University of Texas in 1941 she did not have high expectations, nor did she have any set ideas as to the subjects she wanted to study. It was almost by accident that she came to study mathematics. She went to a large hall full of students on the day for registration:-

... there were few people at the mathematics table so I was sent over there. The man who was sitting there was an old white-haired gentleman. He and I discussed all kinds of things for a long time. When I went to my math class the next day, I found the professor was R L Moore - the same man who talked to me at the registration table.

Mary Ellen had registered for Moore's trigonometry class and she would take one of his classes every year until she graduated with her B.A. degree in 1944. Moore spotted Mary Ellen's mathematical talent right from the start and he was determined that she would become a mathematician:-

I'm a child of Moore. I was always conscious of being manoeuvred by him. I hated being manoeuvred. But part of his technique of teaching was to build your ability to withstand pressure from outside. So he manoeuvred you in order to build your confidence. He built your confidence that you could do anything. I have that total confidence to this day.

As to Moore's teaching methods she wrote:-

His way of teaching was to present you with things that had not yet been proved, and with all kinds of things which might turn out to have a counterexample, and sometimes unsolved problems - that is, unsolved by anyone, not only unsolved by you. So you had some idea of what it meant to be a mathematician - more than the average undergraduate does today.

Although the Moore Method proved good for Mary Ellen Estill, she understood that it was not right for everyone:-

I wouldn't for anything have let my children go to school with Moore! That is, I think that he was destructive to anyone who didn't fit exactly into his pattern, he did not succeed in giving the people that worked with him an education. It's a mistake to go to school under those circumstances in general.

R L Moore was not the only interesting mathematician whom Mary Ellen met at the University of Texas. Much later F Burton Jones who himself had been a student of Moore, had more influence on her mathematics than Moore had had. But as an undergraduate all of her mathematics courses were taught by Moore and she did not specialize in mathematics or any other subject. When she graduated with a BA in 1944 she had not considered going to graduate school in anything. However, when offered a regular Instructorship in mathematics, she began doing topological research under Moore's supervision for her doctorate. She received her PhD in 1949 and Moore arranged an Instructorship for her at Duke University in Durham, North Carolina. She began teaching at Duke in 1949. She found her situation at Duke very different from that at Texas where Moore's students were isolated from all contact with books, other mathematicians, or other mathematical ideas beyond their own. She very much enjoyed reading, and listening to and talking to other mathematicians about their mathematics. In fact, she found her opportunities at Duke so wonderful that [1]:-

... it still seems impossible that anyone would pay me for doing this.

In 1949 Walter Rudin, like Mary Ellen Estill, had just received his PhD in mathematics (from Duke instead of Texas) and he had accepted an Instructorship at Duke for that year also. He left that spring for MIT and in 1952 accepted an Associate Professorship at the University of Rochester. In August of 1953 Mary Ellen and Walter decided to marry and she just arrived at Rochester as his wife. They soon had two daughters, Catherine born in 1954 and Eleanor in 1955. Mary Ellen was a busy wife and mother but already recognized as a successful topologist, one quite actively doing research, and she brought a two year National Science Foundation grant with her. No one in the Rochester mathematics department worked in her field, but the University offered Mary Ellen a Temporary Part time Associate Professorship. For Rochester this title admitted her presence there as a mathematician, avoided any nepotism problem, and got an elementary course taught. For Mary Ellen it gave her legitimate access to the mathematical community and its seminars and library and demanded minimal time. It suited everyone and she held this position from 1953 to 1958. Mary Ellen was not working for the money, but for the love of mathematics [1]:-

I spent more money than most on childcare. It would have been cheaper for me to stay at home.

In 1959 Walter Rudin accepted an appointment as a Full Professor at the University of Wisconsin, Madison. The mathematics department there had a strong group of topologists very interested in Mary Ellen's research and they quickly arranged for her to receive an appointment as a Temporary part time, Lecturer in Mathematics which she enjoyed for 12 years. Mary Ellen and Walter's two sons were born in Madison, Robert Jefferson in 1961 and Charles Michel in 1964. But this appointment not only gave Mary Ellen some time to be a mother which was important to her, it offered enthusiastic and knowledgeable colleagues, research support, PhD students, a private office, and much more.

I was a mathematician, and I always thought of myself as a mathematician. I always had all the goodies that go with being a mathematician. I had graduate students, I had seminars, I had colleagues who loved me. I never had committees. I did lots of mathematics, but I did it because I wanted to do it and enjoyed doing it, not because it would further my career.

In 1971 she was promoted from lecturer to full professor, going in one step from the bottom point of the academic ladder to the top:-

The guilt feelings in the mathematics department were such that nobody even asked me if I wanted to be a professor. I was simply presented with this full professorship.

To begin to understand why the Mathematics Department at the University of Wisconsin should have felt guilty that by 1971 Mary Ellen Rudin was still only a lecturer, we need to take a look at her remarkable mathematical achievements. First, however, let us note that others had recognised her accomplishments long before Wisconsin for, in 1963, the Mathematical Society of the Netherlands awarded her its Prize of *Nieuwe Archief voor Wiskunk*.

In a 1988 interview [3] Rudin explained her mathematical interests:-

... from the beginning, it was the set-theoretic aspects of topology which interested me most. I liked finite and infinite combinatorics. ... I'm basically a problem solver.

To Rudin, mathematics is pattern recognition:-

I draw little pictures and try this thing and that thing. I'm interested in how ideas fit together. Actually I'm very geometric in my thinking. I'm not really interested in numbers.

She began publishing after completing her doctoral thesis. In 1950 the paper *Concerning abstract spaces* was published in the *Duke Mathematical Journal.* It looked at the implications, and relations to various alternatives, of an axiom system for point set theory proposed by Moore in 1932. She continued to look at spaces satisfying a subset of Moore's axioms in *Separation in non-separable spaces* published in 1951. Her 1952 paper *A primitive dispersion set of the plane* provided a positive solution to an unsolved problem contained in R L Wilder's book *Topology of manifolds* (1949). Also in 1952 the paper *Concerning a problem of Souslin's* continued her examination of the implications of Moore's axiom systems, this time motivated by a 1920 problem due to Souslin.

The above papers were all published under her maiden name of Mary Ellen Estill, but beginning with *Countable paracompactness and Souslin's problem* in 1955, she published under her married name of Mary Ellen Rudin. Having looked at some of her earliest papers let us note that she is best known for her ability to construct counter-examples. Perhaps one of the most famous of these example was produced in 1970 when Rudin, using box products, constructed an example of a normal Hausdorff space whose Cartesian product with an interval is not normal.

In August 1974 Rudin gave a series of lectures on set theoretic topology at the CBMS Regional Conference held at the University of Wyoming, Laramie. The notes of these lectures were published by the American Mathematical Society in the following year. In the lectures she surveyed what were then the recent results connecting set theory with the problems of general topology. In particular her many impressive results are put in context in this useful survey.

In 1981 Rudin became the first holder of the Grace Chisholm Young Professorship at Wisconsin. She remained at the University of Wisconsin for the rest of her career, being made professor emeritus. Over the last few years, however, Rudin has produced some very deep mathematical papers. She began publishing a sequence of four papers in 1998 aimed at characterizing the Hausdorff continuous images of compact linearly ordered spaces. These confirm a conjecture by J Nikiel that they are precisely the compact Hausdorff monotonically normal spaces.

Rudin has received many honours for her work, including at least four honorary doctorates, and will continue to receive further awards. She was elected Vice-President of the American Mathematical Society in 1980-81, she has been Governor of the Mathematical Association of America, elected a Fellow of the American Academy of Arts and Science, and elected to the Hungarian Academy of Science. Invited to be the Emmy Noether Lecturer for the Association for Women in Mathematics, she lectured on *Paracompactness.*

When asked how she managed to combine being a full-time mother with being a full-time mathematician, she replied:-

I have never minded doing mathematics lying on the sofa in the middle of the living room with the children climbing all over me. I feel more comfortable and confident when I'm in the middle of things, and to do mathematics you have to feel comfortable and confident.

**Article by:** *J J O'Connor* and *E F Robertson*