Memories of Mary Ellen Rudin · Memories of Mary Ellen Rudin Georgia Benkart, Mirna Džamonja, and...

Memories ofMary Ellen Rudin

Georgia Benkart, Mirna Džamonja, and Judith Roitman,Coordinating Editors

Steven G. Krantz, editor of the Notices, invited the editors of this article to prepare acollective remembrance celebrating Mary Ellen Rudin’s life and work. In doing so, weasked several of Mary Ellen’s colleagues, collaborators, and students to each contribute ashort piece. The choice of contributors was not an easy one, since Mary Ellen had a veryrich mathematical and social life, to which this article can only attest. Space limitationsmeant that we could not accommodate all who would have liked to pay tribute to her,but we hope that they will find themselves, at least to some extent, represented by thereminiscences appearing here.This article is far from being the only initiative to celebrate Mary Ellen Rudin, and weare particularly happy to mention two additional ones: the Mary Ellen Rudin YoungResearcher Award Fund established by Elsevier and an upcoming special issue of Topologyand Its Applications edited by Gary Gruenhage, Jan van Mill, and Peter Nyikos.

Georgia BenkartMary Ellen Estill Rudin, Grace Chisholm Young Pro-fessor Emerita of Mathematics at the University ofWisconsin–Madison and preeminent set-theoretictopologist, died at her home in Madison, Wis-consin, on March 18, 2013. Mary Ellen was bornDecember 7, 1924, in Hillsboro, Texas, south ofDallas–Fort Worth, a small town whose roster ofnotable natives also includes Madge Bellamy, a filmactress of the 1920s and 1930s best known for thehorror movie classic White Zombie.

When she was six, Mary Ellen’s family moved tothe even smaller town of Leakey (LAY-key) situatedin the Texas hill country in a canyon carved outby the Frio and Nueces Rivers. Her mother, IreneShook Estill, had taught high school English. Herfather, Joe Jefferson Estill, was a civil engineer with

Georgia Benkart is professor emerita of mathematics atthe University of Wisconsin-Madison. Her email address [email protected].

Mirna Džamonja is professor of mathematics at the Uni-versity of East Anglia, Norwich, UK. Her email address [email protected].

Judith Roitman is professor emerita of mathematics at theUniversity of Kansas. Her email address is [email protected].

DOI: http://dx.doi.org/10.1090/noti1254

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Mary Ellen’s graduation from the University ofTexas, 1949.

the Texas Highway Department, and the family hadmoved around as his projects dictated until, soonafter their arrival at Leakey, the Depression hit. Asthere was no money to build new roads, the familystayed there throughout Mary Ellen’s school yearswhile her father did surveying work. Her brother,Joe Jefferson Estill Jr., was ten years younger; thetwo siblings always got along famously despite

June/July 2015 Notices of the AMS 617

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Walter’s sister, Vera Usdin, and herhusband, Earl Usdin, at the wedding of

Mary Ellen and Walter in 1953.

their age differ-ence. In thosedays, reachingLeakey from theoutside world re-quired travelingfifty miles upa one-lane dirtroad and ford-ing the Frio Riverseven times. Chil-dren walked toschool or rodethere on horse-back.

In 1941 MaryEllen was one offive graduates inher high schoolclass. That fallshe headed off toAustin to enrollat the University

of Texas and was directed to the table with theshortest line of prospective students, where RobertL. Moore sat registering students for mathemat-ics courses. Impressed with her correct usage of“if/then” and “and/or,” Moore signed Mary Ellenup for a trigonometry course. She was surprised todiscover the next day that he was the professor forthe class. Every single semester during her entireeight years at the University of Texas, from dayone until she graduated with her doctoral degreein 1949, she attended a course taught by Moore[Ke].

Moore’s method of teaching, which eschewedformal lectures, instilled in Mary Ellen a lifelongdeep confidence in solving problems and conjec-tures, though she never used the method in her ownteaching. As she later put it in an interview withAlbers and Reid [AR], at the end of her freshmancalculus course, “I’m not sure I knew the derivativeof sinx was cosx, but I could prove all sorts oftheorems about continuity and differentiabilityand so on!” Mary Ellen often bemoaned the fact thatshe knew so little mathematics and felt cheatedthat she had not been exposed to basic subjectssuch as algebra and analysis.

At Texas she was a member of a terrific cohortof Moore graduate students, including RichardAnderson, Ed Burgess, R. H. Bing, and EdwinMoise, that went on to shape general topology andleave its mark on the mathematical community.In her doctoral thesis, Mary Ellen constructed acounterexample to a well-known conjecture using atechnique now called “building a Pixley-Roy space”after the two mathematicians who later gave amore simplified description of it. At the time

she wrote her thesis, she had never read a singlemathematics paper. She maintained an aversionto reading research papers throughout her career,loved talking about mathematics and sharing herbounty of ideas with anyone who would listen,and was an avid colloquium goer (but always withpencil and paper just in case her interest in thesubject waned).

After Mary Ellen earned her PhD in 1949, Moorereferred her to Duke University, which then had awomen’s college under pressure to hire a femalemathematician. At Duke she met Walter Rudin.Walter had arrived there in 1945 from England,where he had spent the war years in the RoyalNavy after fleeing Austria. Having convinced adean that he should be admitted as a junior eventhough he had never finished high school, fouryears later, in 1949, Walter completed his PhD. Heleft Duke in 1950 for a two-year C. L. E. MooreInstructorship at MIT and following that for theUniversity of Rochester, but Mary Ellen and Waltercontinued to see each other at the winter andsummer Joint Mathematics Meetings. In August1953 they were married at the home of her parentsin Houston. While at Duke, Mary Ellen proved aconjecture of Raymond Wilder, an early Moorestudent, and Wilder applied for an NSF grantto enable her to visit him at the University ofMichigan. When she wired Wilder that she wasgetting married and going to the University ofRochester, somehow he managed for the grant tobe transferred from Michigan to Rochester. TheUniversity of Rochester came up with a part-timeposition for her on short notice, and she continuedher research. Daughter Catherine arrived in July1954, and daughter Eleanor in December 1955,so part-time teaching suited the newly expandedfamily.

The Rudins were on leave from Rochestervisiting Yale for the academic year 1958–59 whenR. H. Bing phoned Walter to invite him to givea summer course at the University of Wisconsin.Walter declined because he already had SloanFellowship funding for the summer but heardhimself asking, “But how about a real job?” [RW].Bing, who was mathematics department chair atthe time, arranged an interview a few weeks later.In 1959 the Rudin family moved to Madison andpurchased a recently built home designed by FrankLloyd Wright that consisted of basically one largeroom with 14-foot ceilings and a huge number ofwindows (rumored to be 137). Sons Robert (Bobby)∗

and Charles (Charlie) were born in Madison in 1961and 1964. Although Walter retreated to the quiet ofhis study to do mathematics, Mary Ellen described

∗Son Robert Rudin died in Madison, Wisconsin, October 13,2014.

618 Notices of the AMS Volume 62, Number 6

her own mode of work as “I lie on the sofa in theliving room with my pencil and paper and thinkand draw little pictures and try this thing andthat thing.…It’s a very easy house to work in. Ithas a living room two stories high, and everythingelse opens onto that.…I have never minded doingmathematics lying on the sofa in the middle of theliving room with the children climbing all over me”[AR].

Madison turned out to be exactly the rightkind of place for them—the right kind of cityand the right kind of mathematics department[RW]. Mary Ellen continued as a part-time lecturerwho supervised graduate students and engagedin essentially full-time research until 1971, when,in Walter’s words [RW], “The anti-nepotism rules,which were actually never a law, had fallen intodisrepute,” and she was promoted from part-timelecturer to full professor. Mary Ellen was somewhatdubious about the professorship because of theteaching and committee work it entailed, but Walterfelt it was best, especially for the children, in casesomething should happen to him. One of my earliestrecollections of Mary Ellen is sharing an elevatorwith her in the mid-1970s, where she showed me aprintout with the small retirement sum she hadaccumulated since coming to Wisconsin, a realitycheck the ever-expanding ranks of adjuncts andpart-timers teaching mathematics today are alsobound to face.

Mary Ellen was known for her extraordinaryability to construct ingenious counterexamples tooutstanding conjectures, many quite complicated.She was the first to construct a Dowker space,thereby disproving a long-standing conjecture ofHugh Dowker that such spaces couldn’t possiblyexist. Years before this noteworthy accomplish-ment, in 1963 her solution to one of the DutchPrize Problems had been recognized by the NieuwArchief voor Wiskunde (the Mathematical Societyof the Netherlands). In later years she was nameda Fellow of the American Academy of Arts andSciences and of the American Mathematical Society,became a member of the Hungarian Academy ofSciences, and received four honorary Doctor ofScience degrees. She gave an invited lecture atthe International Congress of Mathematicians inVancouver in 1974 and the Noether Lecture of theAssociation for Women in Mathematics at the JointMathematics Meetings in 1984. In 1990 she wasawarded the R. H. Bing Prize for significant overallcontributions to mathematics. She served as vicepresident of the AMS 1980–81; as MAA governor1973–75; and on a vast number of AMS, MAA, andnational committees, including the editorial boardsof the Notices and of Topology and Its Applications.Perhaps ironically, she chaired the AMS Committeeon Academic Freedom, Tenure, and Employment

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Eleanor, Charlie, Bobby, and Catherine Rudin.

Security. In her honor, Elsevier established theMary Ellen Rudin Young Researcher Award Fund tocelebrate her achievements and to recognize herlegacy in encouraging talented young researchersin topology. This award was formally announcedat the 47th Spring Topology Meeting a week afterher death, though she had given her approvalof it earlier. The inaugural prize was awarded inSeptember 2013 to Logan C. Hoehn of NipissingUniversity, Canada, a mathematical descendant ofMary Ellen.

Mary Ellen never stopped thinking about math-ematics and working on problems. She was amainstay of the Southern Wisconsin Logic Col-loquium and couldn’t wait to tell me about her“fabulous weekend” when the colloquium hostedthe Association for Symbolic Logic meeting in April2012. The talks were fantastic, she reported; shemet lots of old friends and, of course, participatedin all the social gatherings.

The Rudins welcomed mathematicians, Wrightaficionados, and students from all over the worldto their unique home. The door was always open—literally; it was never locked. As Mary Ellen once saidto me, “People just open the door and say ‘Hello,we’re here.’ ” To which daughter Catherine quicklyadded, “And most of the time we know them.” TheRudins entertained often and hosted everyone witha gracious, relaxed style. I remember a Thanksgivingwhen, with her customary cheerfulness, Mary Ellenannounced to the gathering that dinner would bedelayed, as the turkey, which was being cookedoutside because the oven wasn’t working, had beenon fire.

Mary Ellen was larger than life, and her rep-utation for rescues almost as legendary as her


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Mary Ellen on the sofa.

counterexamples. On more than one occasion shesaved a child from the deep end of a pool or fromcrashing ocean waves. At an AMS summer math-ematics institute in Colorado, a horse suddenlytook off at a gallop, with the young rider whohad slipped from the saddle dangling off to oneside. Mary Ellen charged after them on horseback,corralled the runaway horse, and righted the girl,the daughter of the Rudins’ longtime friend Yalemathematician Shizuo Kakutani, Michiko Kakutani,now a Pulitzer Prize-winning literary critic for theNew York Times.

For well over twenty years, except for the fiveyears that Linda Rothschild was in the department,Mary Ellen and I were the only women mathematicsfaculty members at Wisconsin. Mary Ellen’s passionfor mathematics was contagious, her support andencouragement to all extraordinary. She meantso much to so many; it was inspiring to be apart of her remarkable life. She was so accepting,so warm and welcoming, so genuine, so fiercelyindependent. She would never think to tell anyof us that she had been taken to the hospital byambulance in the middle of the night, except whenit occurred to her she had left the house in just hernightgown with no shoes or money and no way toget back home. After Walter’s death, several of uswould get together regularly for dinner, and shewould want to drive. “We’ll pick you up,” I’d tellher. “Oh, ok,” she would reply. “I’ll be ready”. Andshe always was.

István JuhászMany of us who had close personal and professionalcontact with Mary Ellen Rudin received with deepsorrow the news that she died on March 18 inthe eighty-ninth year of her life. In this age of

István Juhász is professor emeritus at Alfréd Rényi Instituteof Mathematics, Hungarian Academy of Sciences, Budapest,Hungary. His email address is [email protected].

the Wikipedia and numerous other easily availablepublic sources of information, I don’t think it’snecessary for me to give a detailed account ofher great scientific achievements, most of whichwere done in the field of general and set-theoretictopology. Instead, I take this opportunity to recalla few personal memories about her.

The first of these concerns a paper of hersin which she presented an example of a first-countable and CCC regular space that is notseparable and which, as a student, I presentedin our topology seminar. I still remember howproud it made me that I could understand, andmake more or less clear to the audience, her verycomplicated construction. I only much later foundout the explanation of the rather nonstandardnotation and terminology she used (and whichmade her technically very complicated ingeniousconstruction even harder to understand): She wasbrought up in the (in)famous Moore school inTexas, where students were forbidden to studymath books; they had to discover everything ontheir own. I recall a story I heard from Paul Erdos,a good friend of the Rudins, that illustrates this.On one of his visits with them he was surprisedto find out that she did not know what a Hilbertspace was. This was sometime in the 1950s, and,of course, she later became much more familiarwith and well versed in what was going on in herfield.

Our first meeting in person took place at the IMUCongress in Nice in the summer of 1970. Togetherwith my friend and collaborator András Hajnal wewere eager to meet her, and this happened rightafter she arrived in Nice. Her first sentence to uswas “I just proved that there is a Dowker space;”i.e., a normal space whose product with the unitinterval is not normal. To appreciate the weight ofthis sentence, one should know that this meantshe solved the most important open problem ofgeneral topology of the 1960s.

The next year I made a short, basically personal,visit to North America, and she invited me toMadison to give a talk there. That was the first ofmany occasions when she, as a caring host, treatedme so well, both mathematically and socially. Ofcourse, she invited me to their beautiful housedesigned by Frank Lloyd Wright, cooked a greatmeal for me, and showed me around town. I wasglad to be able to return some of her favors inthe summer of 1973 when she came to Hungary,together with her two then-teenage daughters, fora topology conference.

She was the architect of arranging a visitingposition for me at the University of Wisconsin-Madison for the academic year 1974–75. This wasnot a trivial thing in those Cold War times. Oneresult of this visit was our joint paper [JKR] with


Ken Kunen, which is a very frequently cited paperin the field of set-theoretic topology. This visit wasfollowed by many more visits to Madison, mostof them organized by her. But I’d like to mentionanother trip of mine to the US in which she playeda substantial role.

This took place in 1983 when my friend EndreSzemerédi had been diagnosed with a serious caseof cancer while on a visit in the US. Of course, afterhaving found this out, I turned to Mary Ellen to askif she could arrange a visit to Madison for me sothat I could see my ailing friend. It turned out thatthis was not possible because of some financialproblems at UW Madison. Still, she managed toconvince the head of another math departmentin the US—as I later found out, not without somearm twisting—to invite me, thus making it possiblefor me to visit my friend right after he had a veryserious operation. This story has a very happyending: My friend has completely recovered, and,as is well known, in 2012 he was awarded the AbelPrize.

I was fortunate to be present at Mary Ellen’sretirement meeting in Madison in 1991, whereI could witness her efforts trying to help BorisShapirovskii, another great mathematician fightingcancer, unfortunately in his case with no success.

I was very happy and proud when in 1995,mainly thanks to the strong recommendation ofAndrás Hajnal, she was elected to be an honorarymember of the Hungarian Academy of Sciences.Moreover, the next year she and her husband,Walter, after visiting Prague and Vienna, Walter’shometown, managed to come to Budapest so thatshe could deliver her acceptance address at theacademy. A high point of this trip was that myfamily had the good fortune to host a meal in ourhome for the Rudins, the Shelahs, the Hajnals, andPaul Erdos, who, sadly, died just a few weeks later.

I last saw Mary Ellen in person in 2009 at theKunen Fest, Ken Kunen’s retirement celebration.Although quite fragile physically, she was ascheerful as ever. And the last email I got from herwas sent in November of 2010, congratulating uson the occasion of the birth of our grandson.

William FleissnerMary Ellen Rudin was an inspiration to me andother graduate students. Intellectually, she wasenthusiastic about set-theoretic topology. Emo-tionally, she was warm and kind not only tomathematicians but also to their spouses.

I remember seminars on the ninth floor ofVan Vleck Hall, with large windows overlooking

William Fleissner is professor emeritus of mathematics atthe University of Kansas. His email address is [email protected].

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Alex Nagel, Mary Ellen and Walter.

Lake Mendota and the city of Madison. Onceshe presented the Reed-Zenor proof that locallycompact, locally connected normal Moore spacesare metrizable. She drew a Cantor tree, put a unitinterval at the top of each branch, sketched a redcircle about the 0’s and a blue circle about the 1’s.It was not a proof of the theorem. It was morean explanation why a counterexample cannot beconstructed from Martin’s Axiom. Even that wasnot rigorously proved. But she conveyed the ideaswith great enthusiasm. In the audience, I thought,“I want to do mathematics like that!”

There were memorable evenings at their FrankLloyd Wright-designed house on Marinette Trail.(Occasionally, tourists would look in the windows,and Walter would stick out his tongue.) Therewould be food and drink in the kitchen, and acircle of chairs in the living room, which was largein width and length and two stories high. Thewinters in Wisconsin can be bitterly cold, but thenthere would be a fire in the fireplace, making thegathering warm and cheery. The conversation wasvaried, and everyone was welcome to talk, not justthe distinguished professors. The research groupat Wisconsin felt like a big family.

Judith RoitmanShe claimed that she did most of her work on whatshe called her theorem-proving couch, and othermathematicians joked about how they wished theyhad couches which could prove theorems, butit wasn’t just theorems. She created examples—some easy, some difficult, and some which werebreathtaking, almost audacious. Her theorems andexamples cut a wide swath through a world that,when she began, was known as point-set topologyand which relatively soon, in large part because


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Mary Ellen and Paul Erdos when he received anHonorary Doctor of Science degree from the

University of Wisconsin in 1973 (her Erdosnumber is 1).

of the kind of work she did and the kind ofinfluence she had on others, came to be known asset-theoretic topology.

The difficult, breathtaking, audacious examplescame about because, to her, complexity, intrinsiccomplexity, came easily. Her work is, as SteveWatson [W] wrote in the 1993 Festschrift to honorher seventieth birthday, “just hard mathematics,that’s all.” To pick one example: She looked atan arbitrary Suslin tree T and saw not just anL-space out of its branches, which everyone elsesaw, but an S-space constructed out of triplesin ω×ω1 × T whose neighborhoods came fromnot quite subtrees related to each other in anintertwined fashion so complex that, workingthrough it, you think, “This can’t possibly work.”But it does [R72a]. Yet, as is evident in the briefdescription of this construction in her 1975 CBMSnotes [R75], she did not see her S-space from aSuslin tree as particularly difficult. That is what Imean by breathtaking and audacious.

Aside from her results, there was her influence.There were three aspects to this: she broughtpeople together, she encouraged anyone who wasinterested, and she knew where we should belooking: in particular, her 1975 Lectures on SetTheoretic Topology [R75], from the 1974 CBMSRegional Conference in Laramie, set the agenda fordecades.

Her hospitality and warm presence were bothlegendary and a major part of her influence.Consider how I met her. I was a graduate student atBerkeley wanting to use set theory to do topology

and thus wanting to spend time in Madison.Arrangements were made. As soon as I arrived,I called Mary Ellen, per her instructions. Sheapologized that she could not help me get settledright away because her father had just died andshe had to leave town for his funeral. Would I beokay for the next couple of days until she couldget back? I am still astonished at the kindness ofthis gesture from a major mathematician to a verynew graduate student barely past quals.

There were so many gestures like that to so manypeople, helping to form and nurture a community,a floating crew which would meet up in Prague,Warsaw, the Winter School, the Spring TopologyConference.…It was an extraordinarily fertile time.I remember one summer in the mid-1970s whena number of us converged simultaneously onMadison for an impromptu summer-long seminarthat met several times a week. Every meeting beganwith Mary Ellen asking, “Who proved a theoremlast night?” and at every meeting several handswere raised.

Along with her warmth was an immense goodcheer born of deep integrity and an unblinkingsense of reality. When my first baby died ofmeningitis, Mary Ellen wrote the words that Iturned to again and again, telling me how it wasfor her when her son Bobby was born with Downsyndrome, and the doctors laid out a hopelessfuture for him (the hopelessness of which—MaryEllen and Walter being who they were, refusing topay attention to what was then accepted wisdom—did not come to pass): “Your life will never be quitethe same again.” There was tremendous comfortin those words.

Much about Mary Ellen was symbolized to meby the kitchen radio, an AM radio, already very oldwhen I first noticed it. I asked Mary Ellen years laterwhy they didn’t have a newer, better model, andshe said, “Because it still works.” A few years ago Inoticed that the radio was gone. What happened toit? “It stopped working.” This radio was, for me, asymbol of Mary Ellen’s and Walter’s basic decencyand solid values: no matter how many features thenew radios have, you don’t get rid of your old oneif it’s still working.

Mary Ellen was, of course, a woman mathe-matician at a time when there were few womenmathematicians. She belonged, with Julia Robinson,Emma Lehmer and others, to what she called thehousewives’ generation: women who did substan-tial mathematics outside the academy, with onlyoccasional ad hoc positions. I think of those womenas exhibiting enormous strength of character. Ithink they thought of themselves as simply doingmathematics. As F. Burton Jones wrote in theFestschrift volume [J], “Wherever Mary Ellen wasthere was some mathematics.”


Feminist that I was, I would try to engage MaryEllen about how the mathematical communitytreated women, with her as exhibit, if not A, then atleast E or F. She was not interested. “The best way tohelp women in mathematics is to do mathematics!”she roared at me, pounding the breakfast tableat the 1974 Vancouver ICM. Yet she went out ofher way to meet with young women and encouragethem, and she told me many years after Vancouverthat she had come to realize that when young shehad protective blinders: she simply didn’t noticethe differences in how she was treated, so shewasn’t hurt by them.

The last time I saw Mary Ellen was about ayear after Walter died. By then she was using awalker and had moved into the guest bedroomoff the living room to avoid the stairs. But shewas still spending time every day thinking aboutmathematics simply because she loved it so much,working in Walter’s old office on a huge table thatI think was made out of a door plank. We wentthrough the photos and reminiscences that peoplehad sent her in homage to Walter, and her greatno-nonsense good cheerfulness was still there,remembering all the times they had shared. Therewas a call about Bobby’s care, and she excusedherself to deal with it. She was going to meet withfriends for lunch. Her life was full, her affect wasvital, and I could not imagine that this would bethe last time I would see her. But it was.

Mirna DžamonjaMuch has already been said about Mary Ellen’soutstanding personality and human quality. In thisrespect it is impossible for me to separate MaryEllen from her alter ego and husband, Walter Rudin,as this is how I met them and perceived them onesunny day in the summer of 1986, when I wasfinishing the second year of my undergraduatedegree at the University of Sarajevo in Yugoslavia(now Bosnia). They had come to see the universityfrom which Amer Beslagic, one of Mary Ellen’s beststudents, had graduated; their visit was a majorevent in the mathematical life of the city. Theyeach gave an excellent talk, and after Mary Ellen’stalk, I knew I had found myself mathematically: thesubject fascinated me, and the speaker perhapseven more so. I was invited to several socialoccasions with them then, and I was absolutelymesmerized by this couple, so different from eachother, but so inseparable and complementary, theOld World and the New coming together in a uniquemixture, which I had the honour to follow and knowfrom that moment to the end of their lives. I couldnot have imagined then that we would becomesuch close friends and that it is with Walter Rudinthat I would dance at my wedding some years later

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István Juhász, A. Hajnal, Mary Ellen, and MichaelStarbird at the ICM in Vancouver in 1974.

and with Mary Ellen that I would discuss everythingfrom mathematics to cooking. Mary Ellen was abrilliant mathematician, and even though it ishard to stop talking about her as a person, letme stop now and try to say something about hermathematical contribution. This is enormous andI have decided to concentrate on one aspect of itwhich I know the best, Dowker spaces.

The story of Dowker spaces starts withJ. Dieudonné, who in his 1944 article [Di] con-sidered sufficient conditions for a Hausdorfftopological space X to satisfy that for every pair(h, g) of real functions on X with h < g, if h isupper semicontinuous and g is lower semicontin-uous, then there is a continuous f : X → R suchthat h < f < g. C. H. Dowker in his 1951 article[Do] showed that this property is equivalent tothe product X × [0,1] being normal (answeringa conjecture of S. Eilenberg) and left open thepossibility that this was simply equivalent to Xbeing normal. A possible counterexample, so anormal space X for which X × [0,1] is not normal,became known as a Dowker space. In 1955, MaryEllen Rudin [R55] showed that one can obtain aDowker space from a Suslin line. At the time, theconsistency of the existence or nonexistence ofa Suslin line, of course, had not yet been known.M. E. Rudin returned to the problem in 1970 [R71b]when she gave an ingenious ZFC construction ofa Dowker space as a subspace of the product∏n≥1(ωn + 1) in the box topology. This space

has cardinality 2ℵω . This construction of MaryEllen inspired a large amount of research, out ofwhich I mention the two directions that are mostinteresting to me.

The first one is Z. Balogh’s construction [B1] ofa “small Dowker space,” which was in this contexttaken to mean a Dowker space of cardinality2ℵ0 (which, of course, in some universes of set


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Alan Dow, Mary Ellen, and Ivan Reilly at OxfordUniversity.

theory is as large as 2ℵω but in others it is not).Balogh’s method is a very intricate use of certainsequences of functions and elementary submodels,and reading it even today, one feels that itsfull limits have not yet been reached. The otherdirection was taken by M. Kojman and S. Shelah in[KS], where they used pcf theory to find a subspaceof M. E. Rudin’s space which is still Dowker, buthas size ℵω+1. This construction rounds up thelong list of set-theoretic techniques (forcing, largecardinals, pcf, . . .) that were used in Mary Ellen’swork and the work she inspired, carrying the fieldof general topology to what has since becomeknown as set-theoretic topology. The interactionbetween set theory and set-theoretic topology hasbeen most fruitful and has carried both subjectsforward. Mary Ellen was the mother of it all. Letus hope that the unruly children that we are, weshall be able to carry forward not only Mary Ellen’smathematical legacy but also the legacy of thespirit of collaboration, inspiration, mutual respectand joy for all which she was able to create andlive up to.

Franklin TallSet-theoretic topology was inspired by Mary EllenRudin for three decades. Her influence and leader-ship in North America and worldwide were centralto the development of the field.

Mary Ellen grew up in a small town in Texas withno idea that she might become a mathematician.When R. L. Moore was trawling for prospectsat registration at the University of Texas, hesensed she was a good one and began reelingher in. In classic Moore fashion, he created aconfident, powerful researcher who didn’t know

Franklin Tall is professor emeritus of mathematics at theUniversity of Toronto, Toronto, Ontario, Canada. His emailaddress is [email protected].

any mathematics other than what he led her todiscover. Her thesis and first few papers are writtenin Moore’s antiquated language, but Steve Watson[W] has demonstrated that they are still well worthreading. In her thesis [R49], she gave the firstexample of a nonseparable Moore space satisfyingthe countable chain condition.

After positions at Duke (where she met herhusband, Walter Rudin) and Rochester, the Rudinsmoved to the University of Wisconsin at Madison.The UW system had a nepotism regulation thatprevented Mary Ellen from having a regular positionbecause Walter had one. As a result, she stayed a“lecturer,” which had one advantage: she was freeof committee work. In 1971 the regulation wasfinally abolished, and she was promoted directlyto full professor.

I first met Mary Ellen in the summer of 1967. Iwas casting around for a supervisor and Mary Ellenwas very approachable, helpful, and encouraging—qualities that supported a long line of studentsthereafter. I was her first PhD student, and in 1969I had to have the department chair cosign mythesis because, as a lecturer, Mary Ellen was notallowed to be a full-fledged supervisor.

The first paper of Mary Ellen that most set-theoretic topologists have heard of is [R55], inwhich she used a Suslin tree to construct aDowker space. Such a construction would be quiteinteresting if done for the first time now, but in1955, before the consistency of the existence ofsuch trees was known, this was quite extraordinary.She constructed a “real” Dowker space in [R71b],leading to an invitation to address the ICM.

The students of Moore such as Mary Ellen andR. H. Bing were familiar with both elementaryset-theoretic techniques and geometric ones, espe-cially dealing with connected spaces. Mary Ellen’sgeometric abilities proved invaluable in her workon pathological manifolds. Her most noteworthyresult in this area is the consistency (with P. Zenor)and independence of the existence of perfectlynormal nonmetrizable manifolds [R76], [R79].

Some other areas wherein she produced sig-nificant results are: classifying ultrafilters on thenatural numbers (the Rudin-Keisler and Rudin-Frolik orders), showing (with Y. Benyamini andher student Mike Wage) that continuous imagesof Eberlein compacts (weakly compact subspacesof Banach spaces) are also Eberlein [BRW], andestablishing many theorems about normality ofproducts and box products. Let me give just oneexample of a normality theorem, consistently solv-ing a problem of K. Morita. Mary Ellen, with herstudent Amer Beslagic, proved in [BR]: Gödel’sAxiom of Constructibility implies that for a spaceX whose product with every metrizable space isnormal, X is metrizable if and only if X × Z is


normal, for every space Z such that Z’s productswith metrizable spaces are all normal. Amazingly,they proved this by constructing an example!

Mary Ellen’s fame was largely as a producer ofweird and wonderful topological spaces—examplesand counterexamples. She had an uncanny abilityto start off with a space that had some of theproperties she wanted and then push it and pullit until she got exactly what she wanted. In herheyday, several times a week she would receiveinquiries from mathematicians around the world—often nontopologists—asking for an example ofsomething or other. She would answer these bywriting back on the back of the same letter, notfrom “green” sentiments, but to avoid clutter.I am convinced that her avoidance of clutter,especially mentally, was one of the secrets of hermathematical power. By not filling up her memory,she maximized processing capability in her brain.

Mary Ellen’s papers became (and stayed) moreset-theoretic, starting in the late 1960s. There werea variety of reasons for this in addition to whatMarxists might call “historical necessity.” The mostimportant reason was that set theory was beingdone at Wisconsin, most notably by Ken Kunen,who arrived in 1968. David Booth and I wereboth students in logic who had begun thinkingabout applications of set theory to topology; wefrequently went back and forth between Ken andMary Ellen, and that accelerated the collaborationthat made Madison such an exciting place forset-theoretic topology in the 1970s. During thesummers I and many other set-theoretic topologistswould come to visit and interact with Mary Ellenand Ken. The most memorable summer, though,was the one of 1974, when we all went off to theCBMS Regional Conference in Laramie, Wyoming,for Mary Ellen’s lectures on set-theoretic topology[R75]. I remember the sense of excitement we feltat Mary Ellen’s lectures, which set the course ofthe field for the next decade. The resulting book,with chapters on cardinal functions, hereditaryseparability versus hereditary Lindelöfness, boxproducts, and so forth, established a frameworkfor the new field of set-theoretic topology, whichgrew out of general topology but employed newmethods, which led to new questions. It was anexciting time. It was a noteworthy challenge to tryto solve a problem on her problem list.

We celebrated Mary Ellen in 1991 on the occasionof her retirement with a conference in Madison.The proceedings were published in [T1]; much ofthis note is drawn from my contribution [T2] in[T1], where more details of her mathematics canbe found. There is also the excellent biographicalarticle [AR]. Mary Ellen’s research certainly did notstop with her retirement. Particularly impressivewas her complex proof of Nikiel’s Conjecture,

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Mary Ellen and Walter with daughter Catherine’sfamily: Ali Eminov, Catherine, and their sons,Deniz and Adem Rudin.

characterizing the continuous images of compactordered spaces as the compact monotonicallynormal spaces [R01]. As Steve Watson wrote in [W](quoted by Todd Eisworth in his review [E] of [R01]),“Reading the articles of Mary Ellen Rudin, studyingthem until there is no mystery takes hours andhours; but those hours are rewarded, the studentobtains power to which few have access. They arenot hard to read, they are just hard mathematics,that’s all.”

Although her body failed her, her mind remainedsharp as she aged. She stopped traveling, and I wasbusy with family responsibilities, so I saw little ofher in recent years, but she remained and remainsan inspiration for all of us in the field.

Peter Nyikos

I had the honor of delivering a short eulogy forMary Ellen at this year’s Spring Topology andDynamics Conferences (STDC) in Connecticut. Itsaid: “Mary Ellen was a great mathematician. Butshe was much more than that: she was what PrabirRoy called a guru—someone you could turn to foradvice and comfort on all kinds of matters…Tothose of us who knew her well, she was simply‘Mary Ellen.’ Whenever set-theoretic topologistsgot together for a chat, and someone said ‘MaryEllen,’ 99 times out of a hundred, everyone wouldknow who was being talked about.”

I briefly listed some of her main accomplish-ments, including of course the writing of “MaryEllen’s booklet” [R75], another expression that usu-ally gets instant recognition. Among her researchaccomplishments is a beautiful generalization of

Peter Nyikos is professor of mathematics at the University ofSouth Carolina. His email address is [email protected].


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Mary Ellen and and co-author Keiko Chiba inKyoto for the ICM in 1990.

the Hahn-Mazurkiewicz theorem. It is an immedi-ate corollary of her solution to Nikiel’s Conjecture[R01] and of a 1988 theorem of Nikiel [Ni]. The Hahn-Mazurkiewicz theorem states that, if a metrizablespace is a locally connected continuum (compact,connected space), then it is a continuous image of[0,1]. (The converse is elementary.) These spacesare called “Peano continua” in recognition ofPeano’s space-filling curve. Back in 1966 [M], SibeMardešic wrote: “It is natural to ask for a nonmetricanalogue of this theorem.…Recently the interest inthis and related problems has been revived.…” MaryEllen’s solution to Nikiel’s Conjecture allows one tosubstitute “monotonically normal” for “metrizable”in the Hahn-Mazurkiewicz theorem. And the proofthat every metric space is monotonically normal isessentially identical to the usual proof that everymetric space is normal. The search for a naturalextension of the Hahn-Mazurkiewicz theorem tocompact connected, linearly ordered spaces wasakin to the search for a metrization theorem gen-eralizing the Urysohn metrization theorem, and itlasted even longer. (The Bing-Nagata-Smirnov theo-rem was hailed as the solution to this metrizationproblem after a search of over three decades.)

I closed my eulogy by expressing the hope thatthere would be some publications in remembranceof Mary Ellen that would do justice to her greatness,and I am very happy to be able to contribute both

to the special issue of Topology and its Applicationsdedicated to Mary Ellen and to this remembrance.

I got a unique taste of Mary Ellen’s graciousnessand hospitality in early 1974, when I was apostdoctoral student at the University of Chicago.She invited me up to Madison, where I arrived witha bad cold (I naïvely decided not to postpone thevisit, which had already been delayed a number oftimes), but although it was obvious to everyone,she never mentioned it once and had me stayovernight at her house, where I met her two sonsand played board games with them. The sameevening she introduced me to the axioms ♦ and♣ and to Ostaszewski’s S-space, all of which weretotally new to me at the time.

The next two years I saw her at the two STDCconferences, where I became impressed first byher lecturing style and then by the high regard inwhich she was held. There was a panel discussionin Memphis about the future of point-set topology,and the panel included Mary Ellen and other leadingfigures such as R. H. Bing, R. D. Anderson, A. H.Stone, and E. Michael.

Her paper on her screenable Dowker space [R83]solved a 1955 problem of Nagami whether everynormal, screenable space is paracompact [Na]. Theproof of normality was a tour de force, amazing inits originality. I had never seen anything remotelylike it, nor the way she was able to use the intricateset-theoretic axiom ♦++ to define the space itself.To this day I have no idea how it entered intoher mind that a peculiar space like this wouldhave all the properties required to solve Nagami’sproblem nor how she was able to decide on theway to use ♦++ in the definition. One part ofher paper reminded me of an anecdote that wastold about a session in the Laramie workshop.She had been going over a particularly intricateconstruction when F. Burton Jones interrupted:“What allows you to say that?” Mary Ellen replied,“Why that’s—that’s just God-given.” “Yes,” Jonesis supposed to have said, “but what did God saywhen he gave it to you?”

There were some problems in set-theoretictopology which Mary Ellen could not solve but onwhich she did obtain large “consolation prizes.”One such prize was her screenable Dowker space,an offshoot of her unsuccessful attempts to solvea problem for which we still have no consistencyresults: is there a normal space with a σ -disjointbase that is not paracompact? Zoltán Balogh later[B2] came up with one of his “greatest hits”: aZFC example of a screenable Dowker space. MaryEllen’s consistent example is, however, the onlyone known to be collectionwise normal. A recurringtheme in Mary Ellen’s research, up to the veryend of her life, were two further problems aboutDowker spaces: the problem of whether there is a


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Mary Ellen and Walter when they retired in 1991.

normal, linearly Lindelöf space that is not Lindelöfand the problem of whether there is a normal spacewith a σ -disjoint base that is not paracompact.

Kenneth Kunen andArnold Miller

There is an excellent biographical interview ofMary Ellen published in [AR]. Also, there is anentire volume [T1] devoted to articles describingher mathematics. All quotations below are takenfrom [T1].

Mary Ellen Rudin was one of the leading topol-ogists of our time. Besides solving a number ofwell-known outstanding open problems, she was apioneer in the use of set-theoretic tools. She wasone of the first to apply the independence methodsof Cohen and others to produce independenceresults in topology. She did not do forcing argu-ments herself, but many of her papers make use ofset-theoretic statements, such as Martin’s Axiom,Suslin’s Hypothesis, and ♦, that other researchershave shown to be true in some models of set theoryand not in others.

In her thesis [R49] she gave an example ofa nonseparable Moore space that satisfies thecountable chain condition. She published theresults of her thesis in three papers in the DukeMath Journal [R50], [R51], [R52].

To quote Steve Watson, “This cycle repre-sents one of the greatest accomplishments inset-theoretic topology. However the mathematicsin these papers is of such depth that, even forty

Kenneth Kunen is professor emeritus of mathematics at theUniversity of Wisconsin at Madison. His email address [email protected].

Arnold Miller is professor of mathematics at the Univer-sity of Wisconsin-Madison. His email address is [email protected].

years later, they remain impenetrable to all but themost diligent and patient of readers.”

In [R55] she used a Suslin tree to construct aDowker space. The existence of a Suslin tree isconsistent with ZFC but not provable from ZFC.In [R71a, R71b] she constructed a Dowker spacejust in ZFC. This space has cardinality (ℵω)ℵ0 ,whereas the Suslin tree yields a Dowker space ofsize only ℵ1. Also, in 1976 [JKR] she showed thatCH yields a Dowker space of size ℵ1. It is still anopen question whether one can prove in ZFC thatthere is a Dowker space of size ℵ1. There is one ofsize ℵω+1 by Kojman and Shelah [KS].

Her work on Dowker spaces led to an invitedaddress at the International Congress of Mathe-maticians in 1974. It also led to her interest in thebox topology, because her Dowker space of size(ℵω)ℵ0 is a special kind of box product.

Mary Ellen is famous for her work on boxproducts. If Xn (for n ∈ω) are topological spaces,then �nXn denotes the product of the spaces∏n Xn using the box topology ; a base for this

topology is given by all sets∏n Un, where each

Un is open in Xn. The more well-known Tychonovtopology, used to prove the Tychonov Theorem(1935), requires that Un = Xn for all but finitelymany n. Mary Ellen was the first person to proveanything nontrivial about box products. In [R72b]she showed that, assuming CH, �nXn is normal,and in fact paracompact, whenever all the Xn arecompact metric spaces. Also, her 1974 paper [R74]shows that �nXn is paracompact whenever all theXn are successor ordinals; this easily generalizesto the case where the Xn are compact scatteredspaces (see [Ku]). It was later shown by Eric vanDouwen, in ZFC, that there are always compactHausdorff Xn for which �nXn is not normal. Thequestion of which of Mary Ellen’s positive resultscan be proved in ZFC remains an outstandingunsolved problem in topology. Even the case where

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Mary Ellen lecturing.


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At the Kunen Fest 2009: (left to right/front row,back row) Ken Kunen, István Juhász, Mary Ellen,

Jerry Keisler, Arnold Miller, Franklin Tall, JudyRoitman.

all Xn = ω+ 1 remains open, although this case(or when all Xn are compact metric) does followfrom Martin’s Axiom.

Mary Ellen is also famous for her work on βN,the space of ultrafilters on the natural numbers,starting in 1966. She was coinventor of twowell-known partial orders on this space: the Rudin-Keisler order and the Rudin-Frolik order. Thebasic properties of these are given in her 1971paper [R71c], although their historic roots go backsomewhat earlier.

The Rudin-Frolik order led to the first proofin ZFC that the space of nonprincipal ultrafilters,N∗ = βN \N, is not homogeneous. Under CH thiswas already known by a result of Walter Rudin(1956), who proved from CH that there is a P-pointU ∈ N∗ (that is,U is in the interior of every Gδ setcontaining U). Nonhomogeneity follows becauseevery infinite compactum also has a non-P-pointV , and no homeomorphism of N∗ can move thisV to Walter’s U. It was shown much later (Shelah,in the 1970s) that one cannot prove in ZFC thatthese P-points exist.

For the Rudin-Frolik order, for U,V ∈ N∗, saythat U <RF V iff V is a U-limit of some discreteω-sequence of points in N∗. One can show thatthis is a partial order; also, if U <RF V , then nohomeomorphism of N∗ can move V toU, provingnonhomogeneity.

The Rudin-Keisler order comes naturally outof the notion of induced measure. If U,V areultrafilters on any set I, we say that U ≤RK V iffthere is a map f : I → I that induces the measureU from V ; that is, U = {X ⊆ I : f−1(X) ∈ V}.When U,V ∈ N∗, U <RF V implies that U ≤RK Vand V 6≤RK U. This order is of importance inmodel theory, in the theory of ultraproducts, sinceU ≤RK V yields a natural elementary embeddingfrom

∏i Ai/U into

∏i Ai/V . It is also important in

the combinatorics of ultrafilters, since the Rudin-Keisler minimal elements of N∗ are preciselythe Ramsey ultrafilters; such minimal elementsexist under CH but not in ZFC. It is true, butnot completely trivial, that neither of these twoorders is a total order; that is, in ZFC there areU,V ∈ N∗ such that U 6≤RK V and V 6≤RK U (andhence also U 6<RF V and V 6<RF U). The paper[RS] (1979) of Mary Ellen and Saharon Shelah givesthe strongest possible result: for each infiniteκ, there is a family of 22κ ultrafilters on κ thatare pairwise incomparable in the Rudin-Keislerordering. Mary Ellen worked extensively on thequestion of S- and L-spaces. She produced the firstS-space (a hereditarily separable space that is nothereditarily Lindelöf) assuming the existence ofa Suslin tree in 1972 [R72a]. Her Dowker spaceconstructed from CH (see above) is also an S-space. Her 1975 monograph [R75] devotes anentire chapter to S- and L-spaces. To quote StevoTodorcevic, “The terms ‘S-space’ and ‘L-space,’which are predominant in most of the literature onthis subject, are also first found in these lectures.This shows a great influence not only of [R75] butalso of M. E. Rudin’s personality on the generationof mathematicians working in this area, since it israther unusual in mathematics to talk about certainstatements in terms of their counterexamples.”

In 1999, almost a decade after her retirement,Mary Ellen settled a long-standing conjecture inset-theoretic topology by showing that every mono-tonically normal compact space is the continuousimage of a linearly ordered compact space. Thispaper [R01] was the final one in a series of fivewhich gradually settled more and more specialcases of the final result. The construction of thelinearly ordered compact space is extraordinarilycomplex, and to this date there is no simpler proofknown.

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At the time of Walter’s memorial gathering in2010: Mary Ellen with grandson Deniz,

granddaughter Sofie, brother Joe Estill, daughterEleanor, son Charlie, and granddaughter Natalie.


Mary Ellen was by consensus a dominant figurein general topology. Her results are difficult, deep,original, and important. The connections shefound between topology and logic attracted manyset-theorists and logicians to topology. The bestgeneral topologists and set-theorists in the worldpassed regularly through Madison to visit her andwork with her and her students and colleagues.

She had eighteen PhD students, many of whomwent on to have sterling careers of their own. Toquote one of them, Michael Starbird, “From theperspective of a graduate student and collaborator,her most remarkable feature is the flood of ideasthat is constantly bursting from her.. . . It is easyto use the Mary Ellen Rudin model to become agreat advisor. The first step is to have an endlessnumber of great ideas. Then merely give themtotally generously to your students to develop andlearn from. It is really quite simple. For Mary EllenRudin.”

References[AR] D. J. Albers and C. Reid, An interview with Mary

Ellen Rudin, College Math. J. 19:2 (1988), 114–137.www.jstor.org/stable/pdfplus/2686174.pdf+

[B1] Z. T. Balogh, A small Dowker space in ZFC, Proc.Amer. Math. Soc. 124 (1996), no. 8, 2555–2560.

[B2] , A normal screenable nonparacompact spacein ZFC, Proc. Amer. Math. Soc. 126 (1998), no. 6, 1835–1844.

[BR] A. Beslagic and M. E. Rudin, Set-theoretic construc-tions of nonshrinking open covers, Topology Appl.20 (1985), no. 2, 167–177.

[BRW] Y. Benyamini, M. E. Rudin, and M. Wage, Continu-ous images of weakly compact subspaces of Banachspaces, Pacific J. Math. 70 (1977), 309–324.

[Di] J. Dieudonné, Une généralisation des espacescompacts, J. Math. Pures Appl. (9) 23 (1944), 65–76.

[Do] C. H. Dowker, On countably paracompact spaces,Canadian J. Math. 3 (1951), 219–224.

[E] E. T. Eisworth, Review of Rudin, Mary Ellen, Nikiel’sconjecture, Topology Appl. 116 (2001), no. 3, 305–331. MR1857669

[J] F. B. Jones, Some glimpses of the early years, TheWork of Mary Ellen Rudin, ed. Franklin D. Tall, Annalsof the New York Academy of Sciences, 705, 1993, pp.xi–xii.

[JKR] I. Juhász, K. Kunen, and M. E. Rudin, Two morehereditarily separable non-Lindelöf spaces. Canad. J.Math. 28 (1976), 998–1005.

[Ke] P. Kenschaft, Change Is Possible: Stories of Womenand Minorities in Mathematics, Amer. Math. Soc.,Providence, RI, 2005.

[KS] M. Kojman and S. Shelah, A ZFC Dowker space inℵω+1: an application of PCF theory to topology, Proc.Amer. Math. Soc. 126 (1998), no. 8, 2459–2465.

[Ku] K. Kunen, Paracompactness of box products of com-pact spaces, Trans. Amer. Math. Soc. 240 (1978),307–316.

[M] S. Mardešic, On the Hahn-Mazurkiewicz problemin nonmetric spaces, General Topology and Its Rela-tions to Modern Analysis and Algebra II: Proceedings

of the Second Prague Topological Symposium, 1966,J. Novák, ed., Academic Press, 1967, pp. 248–255.

[Na] K. Nagami, Paracompactness and strong screenabil-ity, Nagoya Math. J. 8 (1955), 83–88.

[Ni] J. Nikiel, Images of arcs—a nonseparable version ofthe Hahn-Mazurkiewicz theorem, Fund. Math. 129(1988), no. 2, 91–120.

[R49] M. E. Estill, Concerning Abstract Spaces, PhD Thesis,The University of Texas at Austin, 1949.

[R50] M. E. Rudin, Concerning abstract spaces, Duke Math.J. 17 (1950), 317–327.

[R51] , Separation in non-separable spaces, DukeMath. J. 18 (1951), 623–629.

[R52] , Concerning a problem of Souslin’s, DukeMath. J. 19 (1952), 629–639.

[R55] , Countable paracompactness and Souslin’sproblem, Canad. J. Math. 7 (1955), 543–547.

[R71a] , A normal space X for which X × I is notnormal, Bull. Amer. Math. Soc. 77 (1971), 246.

[R71b] , A normal space X for which X × I is notnormal, Fund. Math. 73 (1971/72), 179–186.

[R71c] , Partial orders on the types in βN, Trans.Amer. Math. Soc. 155 (1971), 353–362.

[R72a] , A normal hereditarily separable non-Lindelöf space, Illinois J. Math. 16 (1972), 621–626.

[R72b] , The box product of countably many com-pact metric spaces, General Topology Appl. 2 (1972),293–298.

[R74] , Countable box products of ordinals, Trans.Amer. Math. Soc. 192 (1974), 121–128.

[R75] , Lectures on Set Theoretic Topology, Expos-itory lectures from the CBMS Regional Conferenceheld at the University of Wyoming, Laramie, Wyo.,August 12–16, 1974. Conference Board of the Math-ematical Sciences Regional Conference Series inMathematics, No. 23, Amer. Math. Soc., Providence,RI, 1975.

[R76] M. E. Rudin and P. Zenor, A perfectly normal non-metrizable manifold, Houston J. Math. 2 (1976), no. 1,129–134.

[R79] M. E. Rudin, The undecidability of the existence of aperfectly normal nonmetrizable manifold, Houston J.Math. 5 (1979), no. 2, 249–252.

[R83] , A normal, screenable, nonparacompact space,Topology Appl. 15 (1983), no. 3, 313–322.

[R01] , Nikiel’s conjecture, Topology Appl. 116(2001), no. 3, 305–331.

[RS] M. E. Rudin and S. Shelah, Unordered types of ultra-filters, Proceedings of the 1978 Topology Conference(Univ. Oklahoma, Norman, Okla., 1978), I. TopologyProc. 3 (1978), no. 1, 199–204 (1979).

[RW] W. Rudin, The Way I Remember It, History of Mathe-matics 12, Amer. Math. Soc., Providence, RI; LondonMath. Soc., London, 1997.

[T1] F. D. Tall, editor, The Work of Mary Ellen Rudin,Papers from the Summer Conference on GeneralTopology and Applications in Honor of Mary EllenRudin Held in Madison, Wisconsin, June 26–29, 1991,Annals of the New York Academy of Sciences, 705,New York Academy of Sciences, New York, 1993.

[T2] Ibid., pp. 1–16.[W] S. Watson, Mary Ellen Rudin’s early work on Suslin

spaces, The Work of Mary Ellen Rudin, F. D. Tall, ed.,Annals of the New York Academy of Sciences, 705,1993, pp. 168–182.


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