Celebratio Mathematica

I had the hon­or of de­liv­er­ing a short eu­logy for Mary El­len at this year’s Spring To­po­logy and Dy­nam­ics Con­fer­ences (STDC) in Con­necti­c­ut. It said: “Mary El­len was a great math­em­atician. But she was much more than that: she was what Prabir Roy called a guru — someone you could turn to for ad­vice and com­fort on all kinds of mat­ters… To those of us who knew her well, she was simply ‘Mary El­len.’ Whenev­er set-the­or­et­ic to­po­lo­gists got to­geth­er for a chat, and someone said “Mary El­len,” 99 times out of a hun­dred, every­one would know who was be­ing talked about.” I briefly lis­ted some of her main ac­com­plish­ments, in­clud­ing of course the writ­ing of “Mary El­len’s book­let” [1]. an­oth­er ex­pres­sion that usu­ally gets in­stant re­cog­ni­tion. Among her re­search ac­com­plish­ments is a beau­ti­ful gen­er­al­iz­a­tion of the Hahn–Mazurkiewicz the­or­em. It is an im­me­di­ate co­rol­lary of her solu­tion to Nikiel’s Con­jec­ture [3] and of a 1988 the­or­em of Nikiel [e3]. The Hahn–Mazurkiewicz the­or­em states that, if a met­riz­able space is a loc­ally con­nec­ted con­tinuum (com­pact, con­nec­ted space), then it is a con­tinu­ous im­age of \( [0, 1] \). (The con­verse is ele­ment­ary.) These spaces are called “Peano con­tinua” in re­cog­ni­tion of Peano’s space-filling curve. Back in 1966 [e2], Sibe Mardeši wrote: “It is nat­ur­al to ask for a non­met­ric ana­logue of this the­or­em….Re­cently the in­terest in this and re­lated prob­lems has been re­vived….” Mary El­len’s solu­tion to Nikiel’s Con­jec­ture al­lows one to sub­sti­tute “mono­ton­ic­ally nor­mal” for “met­riz­able” in the Hahn–Mazurkiewicz the­or­em. And the proof that every met­ric space is mono­ton­ic­ally nor­mal is es­sen­tially identic­al to the usu­al proof that every met­ric space is nor­mal. The search for a nat­ur­al ex­ten­sion of the Hahn–Mazurkiewicz the­or­em to com­pact con­nec­ted, lin­early ordered spaces was akin to the search for a met­riz­a­tion the­or­em gen­er­al­iz­ing the Uryso­hn met­riz­a­tion the­or­em, and it las­ted even longer. (The Bing–Nagata–Smirnov the­or­em was hailed as the solu­tion to this met­riz­a­tion prob­lem after a search of over three dec­ades.)

I closed my eu­logy by ex­press­ing the hope that there would be some pub­lic­a­tions in re­mem­brance of 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 Ap­plic­a­tions ded­ic­ated to Mary El­len and to this re­mem­brance. I got a unique taste of Mary El­len’s gra­cious­ness and hos­pit­al­ity in early 1974, when I was a postdoc­tor­al stu­dent at the Uni­versity of Chica­go. She in­vited me up to Madis­on, where I ar­rived with a bad cold (I naïvely de­cided not to post­pone the vis­it, which had already been delayed a num­ber of times), but al­though it was ob­vi­ous to every­one, she nev­er men­tioned it once and had me stay overnight at her house, where I met her two sons and played board games with them. The same even­ing she in­tro­duced me to the ax­ioms \( \diamond \) and \( \clubsuit \) and to Os­taszewski’s \( S \)-space, all of which were totally new to me at the time. The next two years I saw her at the two STDC con­fer­ences, where I be­came im­pressed first by her lec­tur­ing style and then by the high re­gard in which she was held. There was a pan­el dis­cus­sion in Mem­ph­is about the fu­ture of point-set to­po­logy, and the pan­el in­cluded Mary El­len and oth­er lead­ing fig­ures such as R. H.Bing, R. D. An­der­son, A. H. Stone, and E. Mi­chael.

Her pa­per on her screen­able Dowker space [2] solved a 1955 prob­lem of Nagami wheth­er every nor­mal, screen­able space is para­com­pact [e1]. The proof of nor­mal­ity was a tour de force, amaz­ing in its ori­gin­al­ity. I had nev­er seen any­thing re­motely like it, nor the way she was able to use the in­tric­ate set-the­or­et­ic ax­iom \( \diamond^{++} \) to define the space it­self. To this day I have no idea how it entered in­to her mind that a pe­cu­li­ar space like this would have all the prop­er­ties re­quired to solve Nagami’s prob­lem nor how she was able to de­cide on the way to use \( \diamond^{++} \) in the defin­i­tion. One part of her pa­per re­minded me of an an­ec­dote that was told about a ses­sion in the Lara­m­ie work­shop. She had been go­ing over a par­tic­u­larly in­tric­ate con­struc­tion when F. Bur­ton Jones in­ter­rup­ted: “What al­lows you to say that?” Mary El­len replied, “Why that’s — that’s just God-giv­en.” “Yes,” Jones is sup­posed to have said, “but what did God say when he gave it to you?”

There were some prob­lems in set-the­or­et­ic to­po­logy which Mary El­len could not solve but on which she did ob­tain large “con­sol­a­tion prizes.” One such prize was her screen­able Dowker space, an off­shoot of her un­suc­cess­ful at­tempts to solve a prob­lem for which we still have no con­sist­ency res­ults: is there a nor­mal space with a \( \sigma \)-dis­joint base that is not para­com­pact? Zoltán Ba­logh later [e4] came up with one of his “greatest hits”: a ZFC ex­ample of a screen­able Dowker space. Mary El­len’s con­sist­ent ex­ample is, however, the only one known to be col­lec­tion­wise nor­mal. A re­cur­ring theme in Mary El­len’s re­search, up to the very end of her life, were two fur­ther prob­lems about Dowker spaces: the prob­lem of wheth­er there is a nor­mal, lin­early Lindelöf space that is not Lindelöf and the prob­lem of wheth­er there is a nor­mal space with a \( \sigma \)-dis­joint base that is not para­com­pact.