Celebratio Mathematica

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.