Celebratio Mathematica

Mary Ellen Rudin

A Tribute to Mary Ellen Rudin

by Franklin Tall

Set-the­or­et­ic to­po­logy was in­spired by Mary El­len Rud­in for three dec­ades. Her in­flu­ence and lead­er­ship in North Amer­ica and world­wide were cent­ral to the de­vel­op­ment of the field.

Mary El­len grew up in a small town in Texas with no idea that she might be­come a math­em­atician. When R. L. Moore was trawl­ing for pro­spects at re­gis­tra­tion at the Uni­versity of Texas, he sensed she was a good one and began reel­ing her in. In clas­sic Moore fash­ion, he cre­ated a con­fid­ent, power­ful re­search­er who didn’t know any math­em­at­ics oth­er than what he led her to dis­cov­er. Her thes­is and first few pa­pers are writ­ten in Moore’s an­ti­quated lan­guage, but Steve Wat­son [e4] has demon­strated that they are still well worth read­ing. In her thes­is [1], she gave the first ex­ample of a non­sep­ar­able Moore space sat­is­fy­ing the count­able chain con­di­tion. After po­s­i­tions at Duke (where she met her hus­band, Wal­ter Rud­in) and Rochester, the Rud­ins moved to the Uni­versity of Wis­con­sin at Madis­on. The UW sys­tem had a nepot­ism reg­u­la­tion that pre­ven­ted Mary El­len from hav­ing a reg­u­lar po­s­i­tion be­cause Wal­ter had one. As a res­ult, she stayed a “lec­turer” which had one ad­vant­age: she was free of com­mit­tee work. In 1971 the reg­u­la­tion was fi­nally ab­ol­ished, and she was pro­moted dir­ectly to full pro­fess­or. I first met Mary El­len in the sum­mer of 1967. I was cast­ing around for a su­per­visor and Mary El­len was very ap­proach­able, help­ful, and en­cour­aging — qual­it­ies that sup­por­ted a long line of stu­dents there­after. I was her first PhD stu­dent, and in 1969 I had to have the de­part­ment chair co­sign my thes­is be­cause, as a lec­turer, Mary El­len was not al­lowed to be a full-fledged su­per­visor.

The first pa­per of Mary El­len that most set-the­or­et­ic to­po­lo­gists have heard of is [2], in which she used a Suslin tree to con­struct a Dowker space. Such a con­struc­tion would be quite in­ter­est­ing if done for the first time now, but in 1955, be­fore the con­sist­ency of the ex­ist­ence of such trees was known, this was quite ex­traordin­ary. She con­struc­ted a “real” Dowker space in [3], lead­ing to an in­vit­a­tion to ad­dress the ICM.

The stu­dents of Moore such as Mary El­len and R. H. Bing were fa­mil­i­ar with both ele­ment­ary set-the­or­et­ic tech­niques and geo­met­ric ones, es­pe­cially deal­ing with con­nec­ted spaces. Mary El­len’s geo­met­ric abil­it­ies proved in­valu­able in her work on patho­lo­gic­al man­i­folds. Her most note­worthy res­ult in this area is the con­sist­ency (with P. Zen­or ) and in­de­pend­ence of the ex­ist­ence of per­fectly nor­mal non­met­riz­able man­i­folds [5], [7]. Some oth­er areas wherein she pro­duced sig­ni­fic­ant res­ults are: clas­si­fy­ing ul­tra­fil­ters on the nat­ur­al num­bers (the Rud­in–Keisler and Rud­in–Fro­lik or­ders), show­ing (with Y. Benyamini and her stu­dent Mike Wage) that con­tinu­ous im­ages of Eber­lein com­pacts (weakly com­pact sub­spaces of Banach spaces) are also Eber­lein [6], and es­tab­lish­ing many the­or­ems about nor­mal­ity of products and box products. Let me give just one ex­ample of a nor­mal­ity the­or­em, con­sist­ently solv­ing a prob­lem of K. Mor­ita. Mary El­len, with her stu­dent Amer Bešla­gić, proved in [8]: Gödel’s Ax­iom of Con­struct­ib­il­ity im­plies that for a space \( X \) whose product with every met­riz­able space is nor­mal, \( X \) is met­riz­able if and only if \( X \times Z \) is 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 writ­ing back on the back of the same let­ter, not from “green” sen­ti­ments, but to avoid clut­ter. I am con­vinced that her avoid­ance of clut­ter, es­pe­cially men­tally, was one of the secrets of her math­em­at­ic­al power. By not filling up her memory, she max­im­ized pro­cessing cap­ab­il­ity in her brain.

Mary Ellen and Walter with daughter Catherine’s family: Ali Eminov, Catherine, and their sons, Deniz and Adem Rudin. (Photo courtesy of Ali Eminov.)

Mary El­len’s pa­pers be­came (and stayed) more set-the­or­et­ic, start­ing in the late 1960s. There were a vari­ety of reas­ons for this in ad­di­tion to what Marx­ists might call “his­tor­ic­al ne­ces­sity.” The most im­port­ant reas­on was that set the­ory was be­ing done at Wis­con­sin, most not­ably by Ken Kun­en, who ar­rived in 1968. Dav­id Booth and I were both stu­dents in lo­gic who had be­gun think­ing about ap­plic­a­tions of set the­ory to to­po­logy; we fre­quently went back and forth between Ken and Mary El­len, and that ac­cel­er­ated the col­lab­or­a­tion that made Madis­on such an ex­cit­ing place for set-the­or­et­ic to­po­logy in the 1970s. Dur­ing the sum­mers I and many oth­er set-the­or­et­ic to­po­lo­gists would come to vis­it and in­ter­act with Mary El­len and Ken. The most mem­or­able sum­mer, though, was the one of 1974, when we all went off to the CBMS Re­gion­al Con­fer­ence in Lara­m­ie, Wyom­ing, for Mary El­len’s lec­tures on set-the­or­et­ic to­po­logy [4]. I re­mem­ber the sense of ex­cite­ment we felt at Mary El­len’s lec­tures, which set the course of the field for the next dec­ade. The res­ult­ing book, with chapters on car­din­al func­tions, hered­it­ary separ­ab­il­ity versus hered­it­ary Lindelöfness, box products, and so forth, es­tab­lished a frame­work for the new field of set-the­or­et­ic to­po­logy, which grew out of gen­er­al to­po­logy but em­ployed new meth­ods, which led to new ques­tions. It was an ex­cit­ing time. It was a note­worthy chal­lenge to try to solve a prob­lem on her prob­lem list.

We cel­eb­rated Mary El­len in 1991 on the oc­ca­sion of her re­tire­ment with a con­fer­ence in Madis­on. The pro­ceed­ings were pub­lished in [e2]; much of this note is drawn from my con­tri­bu­tion [e3] in [e2], where more de­tails of her math­em­at­ics can be found. There is also the ex­cel­lent bio­graph­ic­al art­icle [e1]. Mary El­len’s re­search cer­tainly did not stop with her re­tire­ment. Par­tic­u­larly im­press­ive was her com­plex proof of Nikiel’s Con­jec­ture, char­ac­ter­iz­ing the con­tinu­ous im­ages of com­pact ordered spaces as the com­pact mono­ton­ic­ally nor­mal spaces [9]. As Steve Wat­son wrote in [e4] (quoted by Todd Eis­worth in his re­view [e5] of [9]), “Read­ing the art­icles of Mary El­len Rud­in, study­ing them un­til there is no mys­tery takes hours and hours; but those hours are re­war­ded, the stu­dent ob­tains power to which few have ac­cess. They are not hard to read, they are just hard math­em­at­ics, that’s all.”

Al­though her body failed her, her mind re­mained sharp as she aged. She stopped trav­el­ing, and I was busy with fam­ily re­spons­ib­il­it­ies, so I saw little of her in re­cent years, but she re­mained and re­mains an in­spir­a­tion for all of us in the field.


[1] M. E. Es­till: Con­cern­ing ab­stract spaces. Ph.D. thesis, Uni­versity of Texas at Aus­tin, 1949. Ad­vised by R. L. Moore. A con­densed ver­sion was pub­lished in Duke Math. J. 17:4 (1950). MR 2937954 phdthesis

[2] M. E. Rud­in: “Count­able para­com­pact­ness and Souslin’s prob­lem,” Can. J. Math. 7 (February 1955), pp. 543–​547. MR 73155 Zbl 0065.​38002 article

[3] M. E. Rud­in: “A nor­mal space \( X \) for which \( X\times I \) is not nor­mal,” Fund. Math. 73 : 2 (1971–1972), pp. 179–​186. A brief ini­tial ver­sion was pub­lished in Bull. Am. Math. Soc. 77:2 (1971). MR 293583 Zbl 0224.​54019 article

[4] M. E. Rud­in: Lec­tures on set the­or­et­ic to­po­logy (Lara­m­ie, WY, 12–16 Au­gust 1974). CBMS Re­gion­al Con­fer­ence Series in Math­em­at­ics 23. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1975. Re­prin­ted in 1980. MR 367886 Zbl 0318.​54001 book

[5] M. E. Rud­in and P. Zen­or: “A per­fectly nor­mal non­met­riz­able man­i­fold,” Hou­s­ton J. Math. 2 : 1 (1976), pp. 129–​134. MR 394560 Zbl 0315.​54028 article

[6] Y. Benyamini, M. E. Rud­in, and M. Wage: “Con­tinu­ous im­ages of weakly com­pact sub­sets of Banach spaces,” Pac. J. Math. 70 : 2 (1977), pp. 309–​324. MR 625889 Zbl 0374.​46011 article

[7] M. E. Rud­in: “The un­de­cid­ab­il­ity of the ex­ist­ence of a per­fectly nor­mal non­met­riz­able man­i­fold,” Hou­s­ton J. Math. 5 : 2 (1979), pp. 249–​252. MR 546759 Zbl 0418.​03036 article

[8] A. Bešla­gić and M. E. Rud­in: “Set-the­or­et­ic con­struc­tions of non­shrink­ing open cov­ers,” To­po­logy Ap­pl. 20 : 2 (August 1985), pp. 167–​177. MR 800847 Zbl 0574.​54020 article

[9] M. E. Rud­in: “Nikiel’s con­jec­ture,” To­po­logy Ap­pl. 116 : 3 (December 2001), pp. 305–​331. MR 1857669 Zbl 0988.​54022 article