Memories of Mary 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 a collective remembrance celebrating Mary Ellen Rudin’s life and work. In doing so, we asked several of Mary Ellen’s colleagues, collaborators, and students to each contribute a short piece. The choice of contributors was not an easy one, since Mary Ellen had a very rich mathematical and social life, to which this article can only attest. Space limitations meant 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 the reminiscences appearing here. This article is far from being the only initiative to celebrate Mary Ellen Rudin, and we are particularly happy to mention two additional ones: the Mary Ellen Rudin Young Researcher Award Fund established by Elsevier and an upcoming special issue of Topology and Its Applications edited by Gary Gruenhage, Jan van Mill, and Peter Nyikos.

Georgia Benkart

Mary Ellen Estill Rudin, Grace Chisholm Young Pro- fessor Emerita of Mathematics at the University of Wisconsin–Madison and preeminent set-theoretic topologist, died at her home in Madison, Wis- consin, on March 18, 2013. Mary Ellen was born December 7, 1924, in Hillsboro, Texas, south of Dallas–Fort Worth, a small town whose roster of notable natives also includes Madge Bellamy, a ﬁlm actress of the 1920s and 1930s best known for the horror movie classic White Zombie.

When she was six, Mary Ellen’s family moved to courtesy of the Rudin family album. the even smaller town of Leakey (LAY-key) situated in the Texas hill country in a canyon carved out Photo by the Frio and Nueces Rivers. Her mother, Irene Mary Ellen’s graduation from the University of Shook Estill, had taught high school English. Her Texas, 1949. father, Joe Jeﬀerson Estill, was a civil engineer with

Georgia Benkart is professor emerita of mathematics at the Texas Highway Department, and the family had the University of Wisconsin-Madison. Her email address is [email protected]. moved around as his projects dictated until, soon after their arrival at Leakey, the Depression hit. As Mirna Džamonja is professor of mathematics at the Uni- versity of East Anglia, Norwich, UK. Her email address is there was no money to build new roads, the family [email protected]. stayed there throughout Mary Ellen’s school years Judith Roitman is professor emerita of mathematics at the while her father did surveying work. Her brother, University of Kansas. Her email address is [email protected]. Joe Jeﬀerson Estill Jr., was ten years younger; the DOI: http://dx.doi.org/10.1090/noti1254 two siblings always got along famously despite

June/July 2015 Notices of the AMS 617 618

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Mary but for L. Rochester, that C. of following University two-year and a MIT for at Instructorship 1950 He PhD. in his Duke four completed left Walter school, 1949, high in finished later, even years never junior had a as he admitted though be should he that dean a had then which University, Duke to her referred the in interest her case waned). in subject with just always paper listen, (but and goer would pencil colloquium who avid an anyone was with and ideas her of sharing bounty and mathematics about career, talking her loved throughout papers research reading to 2014. described ∗ Ellen Mary mathematics, do of to quiet study the his to retreated 1961 Walter in Although Madison 1964. in and born were (Charlie) Charles and of number huge a and ceilings large 14-foot one with basically room of consisted that Frank Wright by Lloyd designed and home Madison built to recently moved a purchased family later. Rudin weeks the few 1959 a In interview an arranged time, the [ job?” real heard a but about how summer “But asking, the himself for funding give Fellowship Wisconsin. to of him University invite the at to course Walter summer phoned a Bing H. R. expanded newly family. the suited teaching part-time so July in arrived Catherine continued Daughter she research. and notice, her short part-time on a her for with position up The came Rochester. Rochester of to University Michigan of to from grant University transferred the be the for managed was to of he she somehow going University Rochester, that and the Wilder married at wired grant getting him she NSF visit When an to Michigan. her for enable applied Moore to Wilder early an and a Wilder, proved student, Ellen Raymond Mary of Duke, conjecture at While Houston. in ayatrflen uti.Hvn ovne a convinced Having Austria. fleeing after Navy ahmtca.A uesemtWle Rudin. Walter met she Duke At mathematician. ahmtc ae.Semitie naversion an single a maintained read She never paper. had mathematics she thesis, her wrote she ig h a ahmtc eatetcarat chair department mathematics was who Bing, o oetRdnde nMdsn icni,Otbr13, October Wisconsin, Madison, in died Rudin Robert Son fe ayElnere e h n14,Moore 1949, in PhD her earned Ellen Mary After h uiswr nlaefo Rochester from leave on were Rudins The Volume Number 62, RW 6 ∗ ]. her own mode of work as “I lie on the sofa in the living room with my pencil and paper and think and draw little pictures and try this thing and that thing.…It’s a very easy house to work in. It has a living room two stories high, and everything else opens onto that.…I have never minded doing mathematics lying on the sofa in the middle of the living room with the children climbing all over me” [AR]. Madison turned out to be exactly the right kind of place for them—the right kind of city and the right kind of mathematics department [RW]. Mary Ellen continued as a part-time lecturer who supervised graduate students and engaged in essentially full-time research until 1971, when, in Walter’s words [RW], “The anti-nepotism rules, which were actually never a law, had fallen into courtesy of the Rudin family album. disrepute,” and she was promoted from part-time Photo lecturer to full professor. Mary Ellen was somewhat Eleanor, Charlie, Bobby, and Catherine Rudin. dubious about the professorship because of the teaching and committee work it entailed, but Walter felt it was best, especially for the children, in case something should happen to him. One of my earliest Security. In her honor, Elsevier established the recollections of Mary Ellen is sharing an elevator Mary Ellen Rudin Young Researcher Award Fund to with her in the mid-1970s, where she showed me a celebrate her achievements and to recognize her printout with the small retirement sum she had legacy in encouraging talented young researchers accumulated since coming to Wisconsin, a reality in topology. This award was formally announced check the ever-expanding ranks of adjuncts and at the 47th Spring Topology Meeting a week after part-timers teaching mathematics today are also her death, though she had given her approval bound to face. of it earlier. The inaugural prize was awarded in Mary Ellen was known for her extraordinary September 2013 to Logan C. Hoehn of Nipissing ability to construct ingenious counterexamples to University, Canada, a mathematical descendant of outstanding conjectures, many quite complicated. Mary Ellen. She was the first to construct a Dowker space, Mary Ellen never stopped thinking about math- thereby disproving a long-standing conjecture of ematics and working on problems. She was a Hugh Dowker that such spaces couldn’t possibly mainstay of the Southern Wisconsin Logic Col- exist. Years before this noteworthy accomplish- loquium and couldn’t wait to tell me about her ment, in 1963 her solution to one of the Dutch “fabulous weekend” when the colloquium hosted Prize Problems had been recognized by the Nieuw the Association for Symbolic Logic meeting in April Archief voor Wiskunde (the Mathematical Society 2012. The talks were fantastic, she reported; she of the Netherlands). In later years she was named met lots of old friends and, of course, participated a Fellow of the American Academy of Arts and in all the social gatherings. Sciences and of the American Mathematical Society, The Rudins welcomed mathematicians, Wright became a member of the Hungarian Academy of aficionados, and students from all over the world Sciences, and received four honorary Doctor of to their unique home. The door was always open— Science degrees. She gave an invited lecture at literally; it was never locked. As Mary Ellen once said the International Congress of Mathematicians in to me, “People just open the door and say ‘Hello, Vancouver in 1974 and the Noether Lecture of the we’re here.’ ” To which daughter Catherine quickly Association for Women in Mathematics at the Joint added, “And most of the time we know them.” The Mathematics Meetings in 1984. In 1990 she was Rudins entertained often and hosted everyone with awarded the R. H. Bing Prize for significant overall a gracious, relaxed style. I remember a Thanksgiving contributions to mathematics. She served as vice when, with her customary cheerfulness, Mary Ellen president of the AMS 1980–81; as MAA governor announced to the gathering that dinner would be 1973–75; and on a vast number of AMS, MAA, and delayed, as the turkey, which was being cooked national committees, including the editorial boards outside because the oven wasn’t working, had been of the Notices and of Topology and Its Applications. on fire. Perhaps ironically, she chaired the AMS Committee Mary Ellen was larger than life, and her rep- on Academic Freedom, Tenure, and Employment utation for rescues almost as legendary as her

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This visit was followed by many more visits to Madison, most of them organized by her. But I’d like to mention another trip of mine to the US in which she played a substantial role. This took place in 1983 when my friend Endre Szemerédi had been diagnosed with a serious case of cancer while on a visit in the US. Of course, after having found this out, I turned to Mary Ellen to ask if she could arrange a visit to Madison for me so that I could see my ailing friend. It turned out that this was not possible because of some financial problems at UW Madison. Still, she managed to convince the head of another math department in the US—as I later found out, not without some courtesy of Yvonne Nagel.

arm twisting—to invite me, thus making it possible Photo for me to visit my friend right after he had a very Alex Nagel, Mary Ellen and Walter. serious operation. This story has a very happy ending: My friend has completely recovered, and, as is well known, in 2012 he was awarded the Abel Lake Mendota and the city of Madison. Once Prize. she presented the Reed-Zenor proof that locally I was fortunate to be present at Mary Ellen’s compact, locally connected normal Moore spaces retirement meeting in Madison in 1991, where are metrizable. She drew a Cantor tree, put a unit I could witness her efforts trying to help Boris interval at the top of each branch, sketched a red Shapirovskii, another great mathematician fighting circle about the 0’s and a blue circle about the 1’s. cancer, unfortunately in his case with no success. It was not a proof of the theorem. It was more I was very happy and proud when in 1995, an explanation why a counterexample cannot be mainly thanks to the strong recommendation of constructed from Martin’s Axiom. Even that was András Hajnal, she was elected to be an honorary not rigorously proved. But she conveyed the ideas member of the Hungarian Academy of Sciences. with great enthusiasm. In the audience, I thought, Moreover, the next year she and her husband, “I want to do mathematics like that!” Walter, after visiting Prague and Vienna, Walter’s There were memorable evenings at their Frank hometown, managed to come to Budapest so that Lloyd Wright-designed house on Marinette Trail. she could deliver her acceptance address at the (Occasionally, tourists would look in the windows, academy. A high point of this trip was that my and Walter would stick out his tongue.) There family had the good fortune to host a meal in our would be food and drink in the kitchen, and a home for the Rudins, the Shelahs, the Hajnals, and circle of chairs in the living room, which was large Paul Erd˝os,who, sadly, died just a few weeks later. in width and length and two stories high. The I last saw Mary Ellen in person in 2009 at the winters in Wisconsin can be bitterly cold, but then Kunen Fest, Ken Kunen’s retirement celebration. there would be a fire in the fireplace, making the Although quite fragile physically, she was as gathering warm and cheery. The conversation was cheerful as ever. And the last email I got from her varied, and everyone was welcome to talk, not just was sent in November of 2010, congratulating us the distinguished professors. The research group on the occasion of the birth of our grandson. at Wisconsin felt like a big family. William Fleissner Judith Roitman Mary Ellen Rudin was an inspiration to me and She claimed that she did most of her work on what other graduate students. Intellectually, she was she called her theorem-proving couch, and other enthusiastic about set-theoretic topology. Emo- mathematicians joked about how they wished they tionally, she was warm and kind not only to had couches which could prove theorems, but mathematicians but also to their spouses. it wasn’t just theorems. She created examples— I remember seminars on the ninth floor of some easy, some difficult, and some which were Van Vleck Hall, with large windows overlooking breathtaking, almost audacious. Her theorems and William Fleissner is professor emeritus of mathematics at examples cut a wide swath through a world that, the University of Kansas. His email address is fleissn@ku. when she began, was known as point-set topology edu. and which relatively soon, in large part because

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She was not interested. “The best way to help women in mathematics is to do mathematics!” she roared at me, pounding the breakfast table at the 1974 Vancouver ICM. Yet she went out of her way to meet with young women and encourage them, and she told me many years after Vancouver that she had come to realize that when young she had protective blinders: she simply didn’t notice the differences in how she was treated, so she wasn’t hurt by them. courtesy of the Rudin family album. The last time I saw Mary Ellen was about a Photo year after Walter died. By then she was using a István Juhász, A. Hajnal, Mary Ellen, and Michael walker and had moved into the guest bedroom Starbird at the ICM in Vancouver in 1974. off the living room to avoid the stairs. But she was still spending time every day thinking about mathematics simply because she loved it so much, and with Mary Ellen that I would discuss everything working in Walter’s old office on a huge table that I think was made out of a door plank. We went from mathematics to cooking. Mary Ellen was a through the photos and reminiscences that people brilliant mathematician, and even though it is had sent her in homage to Walter, and her great hard to stop talking about her as a person, let no-nonsense good cheerfulness was still there, me stop now and try to say something about her remembering all the times they had shared. There mathematical contribution. This is enormous and was a call about Bobby’s care, and she excused I have decided to concentrate on one aspect of it herself to deal with it. She was going to meet with which I know the best, Dowker spaces. friends for lunch. Her life was full, her affect was The story of Dowker spaces starts with vital, and I could not imagine that this would be J. Dieudonné, who in his 1944 article [Di] con- the last time I would see her. But it was. sidered sufficient conditions for a Hausdorff topological space X to satisfy that for every pair Mirna Džamonja (h, g) of real functions on X with h < g, if h is upper semicontinuous and g is lower semicontin- Much has already been said about Mary Ellen’s uous, then there is a continuous f : X → R such outstanding personality and human quality. In this that h < f < g. C. H. Dowker in his 1951 article respect it is impossible for me to separate Mary [Do] showed that this property is equivalent to Ellen from her alter ego and husband, Walter Rudin, the product X × [0, 1] being normal (answering as this is how I met them and perceived them one a conjecture of S. Eilenberg) and left open the sunny day in the summer of 1986, when I was possibility that this was simply equivalent to X finishing the second year of my undergraduate being normal. A possible counterexample, so a degree at the University of Sarajevo in Yugoslavia normal space X for which X × [0, 1] is not normal, (now Bosnia). They had come to see the university became known as a Dowker space. In 1955, Mary from which Amer Be˘slagić,one of Mary Ellen’s best Ellen Rudin [R55] showed that one can obtain a students, had graduated; their visit was a major Dowker space from a Suslin line. At the time, the event in the mathematical life of the city. They consistency of the existence or nonexistence of each gave an excellent talk, and after Mary Ellen’s a Suslin line, of course, had not yet been known. talk, I knew I had found myself mathematically: the M. E. Rudin returned to the problem in 1970 [R71b] subject fascinated me, and the speaker perhaps when she gave an ingenious ZFC construction of even more so. I was invited to several social a Dowker space as a subspace of the product Q occasions with them then, and I was absolutely n≥1(ωn + 1) in the box topology. This space mesmerized by this couple, so different from each has cardinality 2ℵω . This construction of Mary other, but so inseparable and complementary, the Ellen inspired a large amount of research, out of Old World and the New coming together in a unique which I mention the two directions that are most mixture, which I had the honour to follow and know interesting to me. from that moment to the end of their lives. I could The first one is Z. Balogh’s construction [B1] of not have imagined then that we would become a “small Dowker space,” which was in this context such close friends and that it is with Walter Rudin taken to mean a Dowker space of cardinality that I would dance at my wedding some years later 2ℵ0 (which, of course, in some universes of set

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KS Photo courtesy of the Rudin family album. lnDw ayEln n vnRil tOxford at Reilly Ivan and Ellen, Mary Dow, Alan ayElnge pi ml oni ea with Texas in town small a in up grew Ellen Mary ,weete sdpfter ofidasubspace a find to theory pcf used they where ], [email protected]. ℵ ω+1 hscntuto onsu the up rounds construction This . . . . htwr sdi ayEllen’s Mary in used were that ) ℵ ω u nohr ti not). is it others in but oie fteAMS the of Notices University. lcue, hc a n datg:sewsfree was she advantage: one had which “lecturer,” a eyapocal,hlfl n encouraging— and helpful, Ellen approachable, Mary and very supervisor was a for around casting was hc h sdaSsi ret osrc a construct to tree Suslin a used she which xo fCntutblt mle htfraspace a for that implies Constructibility of Axiom h Wsse a eoimrglto that regulation nepotism a had system UW The 95 eoetecnitnyo h xsec of existence the of consistency the before 1955, hoei oooit aehado s[ is of heard have topologists theoretic hratr a e rtPDsuet n n1969 in and student, students PhD first of her line was long I thereafter. a supported that qualities professor. directly was full promoted to regulation was she the and 1971 abolished, finally In work. committee a of stayed she result, a As one. had Walter because position regular a having from Ellen Mary prevented Madison. at Wisconsin of University the Rudins to the moved Rochester, and Rudin) Walter husband, condition. chain satisfying countable space the Moore nonseparable [ a of thesis example her In reading. Watson Steve [ but language, to antiquated written Moore’s her are papers in led few first he and thesis what Her discover. than other mathematics any X her with Ellen, [ Be˘slagić, Mary in Morita. Amer proved K. student of problem solv- a consistently ing theorem, normality a of example [ Eberlein also are spaces) Banach subspaces of compact images (weakly continuous compacts Eberlein that and of Wage) Benyamini Mike Y. student (with her Rudin- showing and orders), Rudin-Keisler Frolik (the the numbers on natural ultrafilters classifying are: results nificant perfectly [R79]. of [R76], manifolds existence nonmetrizable the normal Zenor) P. of (with independence consistency the and is area this in noteworthy result most Her manifolds. work pathological her on in invaluable proved Ellen’s abilities Mary geometric spaces. elementary connected with both dealing espe- with cially ones, geometric familiar and techniques were set-theoretic Bing H. R. ICM. the address to invitation [ an in to space leading Dowker “real” a extraordinary. constructed quite She was this known, in was but trees such now, time first the for quite done be if would interesting construction a Such space. Dowker supervisor. not my full-fledged was a cosign Ellen be Mary chair to lecturer, allowed department a as the because, have thesis to had I rdcsadbxpout.Ltm iejs one just give me Let products. of box and normality products about theorems many establishing normal, W hs rdc iheeymtial pc is space metrizable every with product whose a eosrtdta hyaesilwl worth well still are they that demonstrated has ] rtmtMr le ntesme f16.I 1967. of summer the in Ellen Mary met first I her met she (where Duke at positions After oeohraesweensepoue sig- produced she wherein areas other Some and Ellen Mary as such Moore of students The set- most that Ellen Mary of paper first The X smtial fadol if only and if metrizable is R49 Volume ,segv h first the gave she ], Number 62, BR BRW :Gödel’s ]: X R55 × R71b ,and ], Z ,in ], is 6 ], normal, for every space Z such that Z’s products with metrizable spaces are all normal. Amazingly, they proved this by constructing an example! Mary Ellen’s fame was largely as a producer of weird and wonderful topological spaces—examples and counterexamples. She had an uncanny ability to start off with a space that had some of the properties she wanted and then push it and pull it until she got exactly what she wanted. In her heyday, several times a week she would receive inquiries from mathematicians around the world— often nontopologists—asking for an example of something or other. She would answer these by courtesy of Ali Eminov. writing back on the back of the same letter, not from “green” sentiments, but to avoid clutter. Photo I am convinced that her avoidance of clutter, Mary Ellen and Walter with daughter Catherine’s especially mentally, was one of the secrets of her family: Ali Eminov, Catherine, and their sons, Deniz and Adem Rudin. mathematical power. By not filling up her memory, she maximized processing capability in her brain. Mary Ellen’s papers became (and stayed) more set-theoretic, starting in the late 1960s. There were characterizing the continuous images of compact a variety of reasons for this in addition to what ordered spaces as the compact monotonically Marxists might call “historical necessity.” The most normal spaces [R01]. As Steve Watson wrote in [W] important reason was that set theory was being (quoted by Todd Eisworth in his review [E] of [R01]), done at Wisconsin, most notably by Ken Kunen, “Reading the articles of Mary Ellen Rudin, studying who arrived in 1968. David Booth and I were them until there is no mystery takes hours and both students in logic who had begun thinking hours; but those hours are rewarded, the student about applications of set theory to topology; we obtains power to which few have access. They are frequently went back and forth between Ken and not hard to read, they are just hard mathematics, Mary Ellen, and that accelerated the collaboration that’s all.” that made Madison such an exciting place for Although her body failed her, her mind remained set-theoretic topology in the 1970s. During the sharp as she aged. She stopped traveling, and I was summers I and many other set-theoretic topologists busy with family responsibilities, so I saw little of would come to visit and interact with Mary Ellen her in recent years, but she remained and remains and Ken. The most memorable summer, though, an inspiration for all of us in the field. was the one of 1974, when we all went off to the CBMS Regional Conference in Laramie, Wyoming, Peter Nyikos for Mary Ellen’s lectures on set-theoretic topology [R75]. I remember the sense of excitement we felt I had the honor of delivering a short eulogy for at Mary Ellen’s lectures, which set the course of Mary Ellen at this year’s Spring Topology and the field for the next decade. The resulting book, Dynamics Conferences (STDC) in Connecticut. It with chapters on cardinal functions, hereditary said: “Mary Ellen was a great mathematician. But separability versus hereditary Lindelöfness, box she was much more than that: she was what Prabir products, and so forth, established a framework Roy called a guru—someone you could turn to for for the new field of set-theoretic topology, which advice and comfort on all kinds of matters…To grew out of general topology but employed new those of us who knew her well, she was simply methods, which led to new questions. It was an ‘Mary Ellen.’ Whenever set-theoretic topologists exciting time. It was a noteworthy challenge to try got together for a chat, and someone said ‘Mary to solve a problem on her problem list. Ellen,’ 99 times out of a hundred, everyone would We celebrated Mary Ellen in 1991 on the occasion know who was being talked about.” of her retirement with a conference in Madison. I briefly listed some of her main accomplish- The proceedings were published in [T1]; much of ments, including of course the writing of “Mary this note is drawn from my contribution [T2] in Ellen’s booklet” [R75], another expression that usu- [T1], where more details of her mathematics can ally gets instant recognition. Among her research be found. There is also the excellent biographical accomplishments is a beautiful generalization of article [AR]. Mary Ellen’s research certainly did not stop with her retirement. Particularly impressive Peter Nyikos is professor of mathematics at the University of was her complex proof of Nikiel’s Conjecture, South Carolina. His email address is [email protected].

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Hahn-Mazurkiewicz the in “metrizable” for normal” to “monotonically one substitute allows Conjecture Nikiel’s to solution Ellen’s Mary revived.…” been has problems related in and interest this the theorem.…Recently this of nonmetric analogue a of for ask to [ recognition natural 1966 is Mardešić in in “It wrote: Back curve. continua” space-filling Peano’s “Peano called are of image continuous a [ is it then (compact, space), continuum connected connected locally a metrizable is a space if that, states theorem Mazurkiewicz Conjecture Nikiel’s to [ solution her of immedi- corollary an ate is It theorem. 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R. as such leading other figures and Ellen topology, Mary included point-set panel of the and future the about Memphis in in regard high the by then and style lecturing her time. the at me to new totally axioms same the The to ♣ them. me introduced with she stay games sons evening two board me her met played had I and and where house, once her at it overnight everyone, mentioned to obvious never was she it although but the times), postpone to not decided a with naïvely arrived (I was I cold where bad Madison, I a to up when me Chicago. invited of 1974, She University the early at student in postdoctoral hospitality and remembrance. this to and Ellen Mary to dedicated of issue special the to okrsae:tepolmo hte hr sa is there whether very of about problem problems the the further to spaces: two Dowker were up life, research, her of Ellen’s end Mary in recurring A theme normal. only collectionwise the be to however, known one is, Mary example space. consistent Dowker Ellen’s screenable a of example ZFC later Balogh Zoltán [ paracompact? not a is with that space base normal consistency a no there have is still results: we which solve for to problem attempts a unsuccessful her space, of Dowker offshoot screenable an her was prize such One on but solve not could Ellen Mary which topology the on decide to able was she Nagami’s how solve nor to problem required properties the all have intricate the use to able was axiom she set-theoretic way remotely the anything nor it, seen like never had I originality. its [ paracompact every is whether space screenable Nagami normal, of problem 1955 a solved Michael. E. and Stone, sspoe ohv ad btwa i o say God did what “but said, have to supposed is ro fnraiywsatu efre mzn in amazing force, de tour a was normality of proof B2 n oOstaszewski’s to and h ettoyasIswhra h w STDC two the at her saw I years two next The graciousness Ellen’s Mary of taste unique a got I hr eesm rbesi set-theoretic in problems some were There [ space Dowker screenable her on paper Her aeu ihoeo i gets is:a hits”: “greatest his of one with up came ] ♦ ++ ntedfiiin n atof part One definition. the in ♦ oooyadisApplications its and Topology ++ S odfietesaeitself. space the define to sae l fwihwere which of all -space, Volume Number 62, σ -disjoint Na ♦ .The ]. 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[email protected]. the at at Wisconsin mathematics of of University emeritus professor is forty Kunen even Kenneth that, depth such of is papers these in mathematics the However topology. set-theoretic Journal Math theory others. set of in models not some and in true be to shown have and Hypothesis, Axiom, Suslin’s Martin’s of as use such make papers statements, argu- her set-theoretic forcing of do many but not herself, independence did ments She produce topology. to in others results and methods Cohen independence the of apply was to She first tools. the of set-theoretic one of use the of in number pioneer a solving Besides time. our of ogists [T1]. from Miller Arnold and Kunen Kenneth space normal a is there whether Lindelöf of not problem the is and that space Lindelöf linearly normal, Mary nievlm [ volume entire nhrtei [ thesis her In topol- leading the of one was Rudin Ellen Mary oqoeSeeWto,“hsccerepre- cycle “This Watson, Steve quote To le n atrwe hyrtrdi 1991. in retired they when Walter and Ellen σ dson aeta sntparacompact. not is that base -disjoint 2015 R0,[5] [R52]. 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Dowker n n n ω ayElni aosfrhrwr nbox on work her for famous is Ellen Mary invited an to led spaces Dowker on work Her [ In soe in open is r ucso rias hsesl generalizes easily this ordinals; successor are R71a X ) ℵ ℵ n le lecturing. Ellen 0 ω+1 R55 n sn the using n saseilkn fbxproduct. box of kind special a is ayElnwstefis esnt prove to person first the was Ellen Mary . , X R71b ℵ n h sdaSsi ret osrc a construct to tree Suslin a used she ] yKja n hlh[KS]. Shelah and Kojman by X 1 lo n17 [ 1976 in Also, . X eoe h rdc ftespaces the of product the denotes n Ku n X n X o which for h osrce okrspace Dowker a constructed she ] n (for ) twsltrsonb rcvan Eric by shown later was It ]). n h oewl-nw Tychonov well-known more The . sprcmatweee l the all whenever paracompact is o topology box n ∈ U X n ω n = r oooia spaces, topological are ) n r opc scattered compact are X X JKR n n Q sntnra.The normal. not is o l u finitely but all for ℵ h hwdthat showed she ] aefrthis for base a ; n 1 hr soeof one is There . U ℵ n n 1 X hr each where , ti tl an still is It . n snormal, is (ℵ X R72b n ω R74 are ) ℵ 0 ] ] , Photo courtesy of the Rudin family album. 627 628 When a led nw yarsl fWle Rudin Walter of result a by known already was Rudin- the space: this on orders partial well-known 15) h rvdfo Hta hr saP-point a is there that CH from proved who (1956), o hnall when (or ftento fidcdmaue If measure. induced of notion the of lrfitr nayset any on ultrafilters trigi 96 h a onetro two of coinventor was She 1966. numbers, natural in the on starting ultrafilters of space the from since ultraproducts, ≤ of U theory the in theory, model and U nonhomogeneity. of homeomorphism ω that that ZFC in prove exist. cannot P-points one these that 1970s) the in V V containing ∈ U ultrafilters, nonprincipal of N space the that ZFC in 1971 her earlier. in somewhat given The are order. [ these paper Rudin-Frolik of properties the basic and order Keisler Axiom. Martin’s from all hr samap a is there vr nnt opcu lohsanon-P-point a has also compactum infinite every hsi ata re;as,if also, order; partial a is this Photo courtesy of the Rudin family album. akrw e ue,Itá uáz ayEllen, Mary Juhász, István Kunen, Ken row) back tteKnnFs 09 lf orgtfotrow, right/front to (left 2009: Fest Kunen the At ∗ sqec fpit in points of -sequence n ohmoopimof homeomorphism no and , h ui-ese re oe aual out naturally comes order Rudin-Keisler The for order, Rudin-Frolik the For proof first the to led order Rudin-Frolik The on work her for famous also is Ellen Mary oWalter’s to from er ese,Anl ilr rnlnTl,Judy Tall, Franklin Miller, Arnold Keisler, Jerry X = RK n U N 6≤ V Q βN U, ∗ = R71c V < i ta is, (that A ω RF V ∈ V \ RK ilsantrleeetr embedding elementary natural a yields i /U N htis, that ; + U V ,atog hi itrcrosg back go roots historic their although ], snthmgnos ne Hthis CH Under homogeneous. not is , U X .Nnooeet olw because follows Nonhomogeneity ). ean pn lhuhti case this although open, remains 1 U N into iff n hsodri fiprac in importance of is order This . f ∗ twssonmc ae (Shelah, later much shown was It . U r opc erc osfollow does metric) compact are : , V I U si h neiro every of interior the in is Q → N sa is < i {X = U ∗ A I RF I i a move can htidcstemeasure the induces that esythat say we , /V U V N lmto oediscrete some of -limit ti loipratin important also is It . ∗ mle that implies n a hwthat show can One . ⊆ U N I oie fteAMS the of Notices ∗ V < U, : a oethis move can RF f to −1 ∈ V ≤ U V U (X) U, hnno then , ≤ U Roitman. proving , RK N V G } V ∈ ∗ V RK δ say , βN are set iff V , . hc rdal ete oeadmr special more and more settled gradually which hc r rdmnn nms fteltrtr on literature the of most in predominant are which ooˇ eic Tetrs‘ Todorˇcević, terms “The hssosagetiflec o nyo [ of only not influence great a shows This ayElnstldaln-tnigcnetr in conjecture long-standing a settled Ellen Mary pc.Hr17 oorp [ monograph 1975 Her space. ae ftefia eut h osrcino the of construction The result. final the of cases resi oa re;ta s nZCteeare there ZFC in but is, true, that order; is two total It these a of ZFC. is neither orders in that not trivial, completely but not CH elements under minimal such exist ultrafilters; Ramsey the niecatrto chapter entire [ 1972 in tree of Suslin existence a the assuming Lindelöf) hereditarily S of question infinite each for result: κ possible strongest the [ also hence U, of elements minimal Rudin- the Keisler since ultrafilters, of combinatorics the ope,adt hsdt hr sn ipe proof simpler no is there known. date this to and extraordinarily complex, is space compact ordered linearly This space. compact [ ordered paper linearly a continuous of the image is space compact normal monotonically every that showing by topology set-theoretic counterexamples.” their of terms in certain about statements is talk it to mathematics since in area, unusual rather this generation in the working on mathematicians personality of Rudin’s E. M. of also lectures. these in found first also are subject, this reig ayElnwre xesvl nthe on extensively worked Rudin-Keisler Ellen the Mary in ordering. incomparable pairwise are osrce rmC seaoe sas an also is above) (see CH from constructed rnduhe oe rte o sil daughter Estill, Joe brother Sofie, granddaughter lao,snCale n rnduhe Natalie. granddaughter and Charlie, son Eleanor, Photo courtesy of Scott Gaff. RS sae( eeiaiysprbesaeta snot is that space separable hereditarily (a -space hr safml f2 of family a is there , n19,ams eaeatrhrretirement, her after decade a almost 1999, In ∈ V 17)o ayElnadShrnSea gives Shelah Saharon and Ellen Mary of (1979) ] At h ieo atrsmmra ahrn in gathering memorial Walter’s of time the R01 N ∗ 00 ayElnwt rnsnDeniz, grandson with Ellen Mary 2010: a h nloei eiso five of series a in one final the was ] uhthat such S 6< U and - RF S L and - V sae.Sepoue h first the produced She -spaces. 6≤ U and R72a L 2 Volume RK sae.T ut Stevo quote To -spaces. κ S 6< V sae n ‘ and -space’ lrfitr on ultrafilters V .HrDwe space Dowker Her ]. and N RF R75 ∗ U Number 62, 6≤ V r precisely are eoe an devotes ] .Tepaper The ). RK L R75 -space,’ U κ but ] (and that S 6 - Mary Ellen was by consensus a dominant figure of the Second Prague Topological Symposium, 1966, in general topology. Her results are difficult, deep, J. Novák, ed., Academic Press, 1967, pp. 248–255. original, and important. The connections she [Na] K. Nagami, Paracompactness and strong screenabil- ity, Nagoya Math. J. 8 (1955), 83–88. found between topology and logic attracted many [Ni] J. Nikiel, Images of arcs—a nonseparable version of set-theorists and logicians to topology. The best the Hahn-Mazurkiewicz theorem, Fund. Math. 129 general topologists and set-theorists in the world (1988), no. 2, 91–120. passed regularly through Madison to visit her and [R49] M. E. Estill, Concerning Abstract Spaces, PhD Thesis, work with her and her students and colleagues. The University of Texas at Austin, 1949. She had eighteen PhD students, many of whom [R50] M. E. Rudin, Concerning abstract spaces, Duke Math. J. 17 (1950), 317–327. went on to have sterling careers of their own. To [R51] , Separation in non-separable spaces, Duke quote one of them, Michael Starbird, “From the Math. J. 18 (1951), 623–629. perspective of a graduate student and collaborator, [R52] , Concerning a problem of Souslin’s, Duke her most remarkable feature is the flood of ideas Math. J. 19 (1952), 629–639. that is constantly bursting from her.... It is easy [R55] , Countable paracompactness and Souslin’s problem, Canad. J. Math. 7 (1955), 543–547. to use the Mary Ellen Rudin model to become a [R71a] , A normal space X for which X × I is not great advisor. The first step is to have an endless normal, Bull. Amer. Math. Soc. 77 (1971), 246. number of great ideas. Then merely give them [R71b] , A normal space X for which X × I is not totally generously to your students to develop and normal, Fund. Math. 73 (1971/72), 179–186. learn from. It is really quite simple. For Mary Ellen [R71c] , Partial orders on the types in βN, Trans. Amer. Math. Soc. 155 (1971), 353–362. Rudin.” [R72a] , A normal hereditarily separable non- Lindelöf space, Illinois J. Math. 16 (1972), 621– References 626. [AR] D. J. Albers and C. Reid, An interview with Mary [R72b] , The box product of countably many com- Ellen Rudin, College Math. J. 19:2 (1988), 114–137. pact metric spaces, General Topology Appl. 2 (1972), www.jstor.org/stable/pdfplus/2686174.pdf+ 293–298. [B1] Z. T. Balogh, A small Dowker space in ZFC, Proc. [R74] , Countable box products of ordinals, Trans. Amer. Math. Soc. 124 (1996), no. 8, 2555–2560. Amer. Math. Soc. 192 (1974), 121–128. [B2] , A normal screenable nonparacompact space [R75] , Lectures on Set Theoretic Topology, Expos- in ZFC, Proc. Amer. Math. Soc. 126 (1998), no. 6, 1835– itory lectures from the CBMS Regional Conference 1844. held at the University of Wyoming, Laramie, Wyo., [BR] A. Be˘slagić and M. E. Rudin, Set-theoretic construc- August 12–16, 1974. Conference Board of the Math- tions of nonshrinking open covers, Topology Appl. ematical Sciences Regional Conference Series in 20 (1985), no. 2, 167–177. Mathematics, No. 23, Amer. Math. Soc., Providence, [BRW] Y. Benyamini, M. E. Rudin, and M. Wage, Continu- RI, 1975. ous images of weakly compact subspaces of Banach [R76] M. E. Rudin and P. Zenor, A perfectly normal non- spaces, Pacific J. Math. 70 (1977), 309–324. metrizable manifold, Houston J. Math. 2 (1976), no. 1, [Di] J. Dieudonné, Une généralisation des espaces 129–134. compacts, J. Math. Pures Appl. (9) 23 (1944), 65–76. [R79] M. E. Rudin, The undecidability of the existence of a [Do] C. H. Dowker, On countably paracompact spaces, perfectly normal nonmetrizable manifold, Houston J. Canadian J. Math. 3 (1951), 219–224. Math. 5 (1979), no. 2, 249–252. [E] E. T. Eisworth, Review of Rudin, Mary Ellen, Nikiel’s [R83] , A normal, screenable, nonparacompact space, conjecture, Topology Appl. 116 (2001), no. 3, 305– Topology Appl. 15 (1983), no. 3, 313–322. 331. MR1857669 [R01] , Nikiel’s conjecture, Topology Appl. 116 [J] F. B. Jones, Some glimpses of the early years, The (2001), no. 3, 305–331. Work of Mary Ellen Rudin, ed. Franklin D. Tall, Annals [RS] M. E. Rudin and S. Shelah, Unordered types of ultra- of the New York Academy of Sciences, 705, 1993, pp. filters, Proceedings of the 1978 Topology Conference xi–xii. (Univ. Oklahoma, Norman, Okla., 1978), I. Topology [JKR] I. Juhász, K. Kunen, and M. E. Rudin, Two more Proc. 3 (1978), no. 1, 199–204 (1979). hereditarily separable non-Lindelöf spaces. Canad. J. [RW] W. Rudin, The Way I Remember It, History of Mathe- Math. 28 (1976), 998–1005. matics 12, Amer. Math. Soc., Providence, RI; London [Ke] P. Kenschaft, Change Is Possible: Stories of Women Math. Soc., London, 1997. and Minorities in Mathematics, Amer. Math. Soc., [T1] F. D. Tall, editor, The Work of Mary Ellen Rudin, Providence, RI, 2005. Papers from the Summer Conference on General [KS] M. Kojman and S. Shelah, A ZFC Dowker space in Topology and Applications in Honor of Mary Ellen ℵω+1: an application of PCF theory to topology, Proc. Rudin Held in Madison, Wisconsin, June 26–29, 1991, Amer. Math. 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