#### by Michael Starbird

I entered the University of Wisconsin, Madison in the spring of 1971 taking two math courses, one a grad course and the other an undergraduate point-set topology class taught by Mary Ellen Rudin. I had taken an earlier course in topology at Pomona College, but it wasn’t clear to me how much I had learned as I hadn’t been a serious student as an undergraduate.

In any case, Mary Ellen was an interesting teacher. She did not use the
Moore method, but on the other hand she didn’t prepare her lectures. So
the effect was that she would come to class and attempt to prove a
theorem that would come to her mind, but she wasn’t always successful
at finding a proof. Her teaching was not precise. She would say, “Let __\( X \)__
be a Hausdorff space.” Then she would write __\( Y \)__, and she really meant __\( Z \)__,
and the correct thing would have been __\( W \)__. She would begin and then get
stuck. Well, for me this situation was great, because I would try to
help. As I recall it, after she would get stuck, I would chime in with
some suggestions not necessarily beneficial to the cause. It was great
fun to interact with her in this way. Her approach to teaching would not
provide a role model for most teachers. She wasn’t really that great a
teacher in any traditional sense, but she was definitely enthusiastic,
and for me it was a very positive and interactive experience. I don’t
remember how much I learned in that class, but it was definitively
entertaining and engaging.

The next semester I took the standard graduate-level introductory point-set topology course that was taught using the Moore method, and that’s where I really took off as a student because I loved it. That course was taught by Russ McMillan, who was a student of R H Bing, who was a student of R. L. Moore. I just loved that class. It was like a collection of puzzles, and I found them delightful. For the first time in my life, I actually understood the whole collection of mathematics from a course. At the end of that year, you could give me blank pieces of paper, and I could write down and prove everything from the course for the whole year. It was a totally different level of understanding for me than I had ever before experienced. In any case, I was quite successful at proving the theorems in point-set topology. In addition to courses, there were topology seminars, which I attended, and so I got to know the topology faculty members, including Mary Ellen Rudin.

The University of Wisconsin Department of Mathematics, housed in Van Vleck Hall, exuded a positive and welcoming spirit. The top (ninth) floor consisted of a lounge surrounded by windows that overlooked the city and lakes. There were blackboards around the room and at most times during the day graduate students and faculty members would be cheerfully interacting, discussing mathematics at the boards or at the tables or playing chess or just talking. It was a really lively and fun environment for both the professors and the graduate students. The department certainly had created an atmosphere where it was easy to get to know the faculty members and to interact with them.

Mary Ellen Rudin became my thesis advisor. She described the process of me connecting with her by saying that she took me on by force of personality. I don’t recall any decision-making on my part being involved in the process.

In any case, her method of advising was to give me a few unsolved problems on a piece of paper. I was able to prove some of them and that was my dissertation. So I finished in three and a half years. In the meantime, R H Bing had been offered a job at the University of Texas, so he left in 1973. When he left he said, “When you finish, come to Texas.” So I did, and I’m still here. So I’ve never applied for a job.

Mary Ellen was known for proving incredibly arcane things, including extremely complicated set-theoretic examples. She was famous for having produced a topological space that was normal but whose product with an interval was not. Her example is so complicated that it is difficult to imagine a big part of what is going on and why.

My dissertation was about normality in product spaces. Mary Ellen and I worked on several theorems about product spaces and normality of products with various conditions on the two factors. She and I wrote several papers together about product spaces.

It was just great fun to work with Mary Ellen, both as a graduate student and afterwards. First of all, she was completely generous with her ideas. She would often come up with some crazy-sounding idea and after working on it for some time, I would realize, to my amazement, that it worked. I recall one idea that I thought was completely crazy. It involved randomly selecting two points from each open set in a collection of open sets in the product space. I thought this strategy was one that a rank beginner would use, and it could not lead anywhere — but, of course, it did. She was astonishingly good at dealing with abstract sets. She had an overflow of ideas, all of which she freely distributed to me and others who worked with her.

After I left Wisconsin, we also wrote a paper about the normal Moore space
conjecture, which at that time was unsettled.
A *Moore space* __\( X \)__ is a topological space such that for any closed set __\( C \)__ and any point __\( x \)__ in the complement of __\( C \)__, these two conditions hold: (1) __\( C \)__ and __\( x \)__ are separated by two open sets (__\( X \)__ is regular Hausdorff), and (2) there is a countable collection of covers of __\( X \)__ such that for one of these covers, every open set containing __\( x \)__ is disjoint from __\( C \)__.

Moore spaces have some of the properties of metric spaces, but they are not necessarily normal. The Normal Moore Space Problem asks whether a Moore space that is also a normal space must then be a metrizable space.

People had tried to settle this question for many years. Bing showed that __\( X \)__ is metrizable under a stronger version of normality. In the 1930s,
F. B. Jones
proved that under the assumption of the continuum hypothesis, separable
normal Moore spaces are metrizable. So in the mid 1970s, the normal
Moore space conjecture sat in this unsettled state.

Mary Ellen and I turned our minds to this question and wrote a paper. Our paper constructed a family of spaces with the property that if there were a counterexample of a likely type (which we defined in the paper) to the normal Moore space conjecture, then one of the spaces we constructed would be an example. It had many of Mary Ellen’s characteristic features, e.g., complexity and lots of subscripts.

However we were doomed to failure in our attempts to prove the conjecture because soon after, Peter Nyikos showed that the normal Moore space conjecture is actually independent of the standard axioms of set theory. So our paper about the normal Moore space problem was not of much use. But it was certainly a joy to write with Mary Ellen.

Let me recall some of what I know about Mary Ellen’s life before I knew her.

Mary Ellen told me about her very first experiences at the University of Texas. She came from a very tiny town. I recall her telling me that she had to ride a horse to get out of town. In any case, she arrived at UT when she was 16 or so in the early 1940s. At that time, the way registration worked for mathematics classes is that R. L. Moore would sit at a table in Gregory Gym, and the students would come up to his desk to be put into classes. Moore would ask each student some questions that he had devised to determine how bright the student was. If Moore decided the person was mathematically talented, then he would put the student in his own class. If the student was not so good, he enrolled the student in someone else’s class.

When Mary Ellen arrived at his desk, she demonstrated her brilliance very clearly. Not only did Moore choose her for his class, but, in fact, he taught the class (which was trigonometry) even though he wasn’t scheduled to teach it. That is, Moore taught that extra course just so he could have her in a class.

Mary Ellen certainly had a university mathematics experience dominated by R. L. Moore. He taught every one of the mathematics classes she took from her freshman year through the end of graduate school except for three classes, and Moore directed her Ph.D. thesis. Mary Ellen did not think of herself as a person who was destined from the start to become a mathematician. She suspected that she might have become a linguist or something else had it not been for the influence of Moore.

As we all know, Moore’s method of instruction involved asking the students to discover the mathematics on their own. So Mary Ellen had many years of practice in proving theorems on her own during her undergraduate and graduate school years at UT. She told me that there is no question she became a mathematician owing to Moore’s influence; however, she was very critical about not having received a broad enough mathematical education. Moore gave her a very narrow vision of set-theoretic topology, and she was not exposed to whole areas of mathematics. One consequence of this education was that the problems she worked on typically tended to be old-fashioned point-set topology problems rather than questions that involved more up-to-date mathematics. I do not know when she learned the set-theoretic logic that she employed and explored. I don’t think Moore presented that kind of set theory in his classes, but I’m not sure.

As a student, Mary Ellen was often in small classes with Moore where there were perhaps two or three other students who were like herself in being much stronger than the others. She once wondered aloud to me whether the weaker students got much out of those classes since the strong students would compete and dominate the class.

Another memory she had of Moore’s classes is what he would do if the students had not proved any theorems that day. In that case, Moore would talk about politics. His views on the subject were sufficiently distasteful that Mary Ellen said the threat of having to listen to him talk about politics was a great incentive for the students to prove theorems and so fill the class time with mathematics.

Mary Ellen lived with her husband Walter and their family in a Frank Lloyd Wright house in Madison. The architect was active there and had built several houses in the area. (I believe the Rudins were the only owners of their house.) The house had a sort of boxy design: its open plan featured a two-story living room in the middle (whose vertical reach was terminated only by the roof) and a big fireplace. Around the living room on the second floor was a walkway, with sliding doors that opened onto the upstairs bedrooms. So you could see or access most of the rooms in the house from the living room. Mary Ellen created a chaotic and joyful environment for her family and her many guests in this space.

Mary Ellen had four children who were all grown or in college by the time I met her.1 Mary Ellen was such a loud, lively person, her household must have been full of life and laughter, especially when her children were young. She was not bothered by noise or distractions, and was able to do her arcane, complex mathematics while sitting on the couch in her living room while her children romped around her, probably yelling and screaming and jumping on her with regularity. She also did mathematics in the car (presumably when someone else was driving). Her ability to shut out distractions was remarkable. Her husband, Walter, had a different style: He would retreat to his study to enjoy privacy and quiet when he worked.

Walter was an amazing mathematician. Here is one story about
him. One
time Walter came with Mary Ellen as a spouse to a conference on
set-theoretic topology. To pass the time, he attended some of the talks. At
one of these talks the lecturer presented a big unsolved problem about the
Stone–Čech
compactification of the natural numbers. This space is called
__\( \beta \mathbb{N} \)__. The question had to do with the limit points of __\( \beta \mathbb{N} \)__. So Walter went
home and settled the question. It was not his field at all and yet he
was able to solve one of the challenging problems in the field.

Mary Ellen had been at the University of Wisconsin for many years when
I arrived in 1971, but she had had very few Ph.D. students during that
time because of nepotism rules. When she and Walter arrived in Wisconsin,
Walter was a professor, and because of being married to
him, Mary Ellen was barred from having a professorial rank. Instead,
she was a lecturer and lecturers were not allowed to direct Ph.D.
students. In the late 1960s, Mary Ellen was acknowledged to be the
best set-theoretic topologist in the world. I don’t really know the
administrative history of their change of heart, but in the late 1960s
the University of Wisconsin decided that *full professor* was the
more appropriate rank for Mary Ellen, so she skipped a few steps and
jumped directly from being a lecturer to being a full professor. Then
she started taking on Ph.D. students.

I arrived at Madison soon after Mary Ellen became a professor, and when I finished in 1974, I was only her fourth Ph.D. student. Her first student was Frank Tall, who graduated in 1969.

One of the delightful and inspiring features of Mary Ellen was the kindness with which she treated everyone. I remember several instances of her matchless generosity of spirit. One incident that illustrated her character I remember particularly well, because I am still not certain what went on in her mind.

As you may know, not all mathematicians are models of civility and grace. One day I recall being in a conversation that involved Mary Ellen, a certain rather rude and obnoxious mathematician (whom I will not name) and myself. This obnoxious guy would say impressively rude things, and Mary Ellen would cheerfully respond exactly as if he had spoken in the most civil way imaginable. I witnessed this type of conversational interaction on several occasions. I always wondered whether Mary Ellen was simply oblivious to obnoxiousness or whether she knew the guy was obnoxious, but just decided not to acknowledge it. We will never know, but it certainly exemplified to me a model of kindness.

The last time I saw Mary Ellen was just the year before she died. There was a summer conference in Madison, and I had gone to it and took the trouble to go to her Frank Lloyd Wright house to visit her. Walter had died several years before and one thing she told me about was the last years of his life. I had seen him in about 2000 when his Parkinson’s had already started to kick in. His symptoms gradually became worse and worse, and Mary Ellen’s description of the problems he had during his last years was poignant, for he was prone to falling. She told me about taking a walk with him when he fell down and tore his scalp. It was such a sad story, yet when I saw her that summer, she was her usual cheerful self. She showed me her photo album and told me about many events in her long and wonderful life. She was about 88 then, and was using a walker, but her spirit was indomitable.

After my visit, when it was time to go, she offered to drive me back to the convention center. I had taken a cab to get there, and was a bit surprised that she was still driving, but she assured me that it would be fine. Her driving was sound… most of the time: It did involve going a couple of blocks the wrong way on a one-way road, but perhaps the one-way arrow was just intended to be a suggestion.

For me, Mary Ellen Rudin will always be an inspiration as a mathematician
and as a human being. From the perspective of a graduate student and
collaborator, perhaps her most remarkable quality was the flood of ideas
that constantly burst from her. It is easy to use the Mary Ellen Rudin
model to become a great advisor and mathematician. The first step is
to have an endless number of great ideas. Then merely give them
generously and completely to your students and colleagues. It
is really quite simple. *If you are Mary Ellen Rudin.*

Following her model for becoming a better human being is equally simple. Just have an endless capacity for verve and joy. Discover the good in everyone and find a way to fail to see the defects in others. Then welcome everyone into a world of vibrant curiosity and uplifting good cheer! Mary Ellen Rudin was a paragon in all these things.