# Celebratio Mathematica

## Mary Ellen Rudin

### A Tribute to Mary Ellen Rudin

#### by Mirna Džamonja

Much has already been said about Mary El­len’s out­stand­ing per­son­al­ity and hu­man qual­ity. In this re­spect it is im­possible for me to sep­ar­ate Mary El­len from her al­ter ego and hus­band, Wal­ter Rud­in, as this is how I met them and per­ceived them one sunny day in the sum­mer of 1986, when I was fin­ish­ing the second year of my un­der­gradu­ate de­gree at the Uni­versity of Sa­ra­jevo in Yugoslavia (now Bos­nia). They had come to see the uni­versity from which Amer Bešla­gić, one of Mary El­len’s best stu­dents, had gradu­ated; their vis­it was a ma­jor event in the math­em­at­ic­al life of the city. They each gave an ex­cel­lent talk, and after Mary El­len’s talk, I knew I had found my­self math­em­at­ic­ally: the sub­ject fas­cin­ated me, and the speak­er per­haps even more so. I was in­vited to sev­er­al so­cial oc­ca­sions with them then, and I was ab­so­lutely mes­mer­ized by this couple, so dif­fer­ent from each oth­er, but so in­sep­ar­able and com­ple­ment­ary, the Old World and the New com­ing to­geth­er in a unique mix­ture, which I had the hon­our to fol­low and know from that mo­ment to the end of their lives. I could not have ima­gined then that we would be­come such close friends and that it is with Wal­ter Rud­in that I would dance at my wed­ding some years later and with Mary El­len that I would dis­cuss everything from math­em­at­ics to cook­ing. Mary El­len was a bril­liant math­em­atician, and even though it is hard to stop talk­ing about her as a per­son, let me stop now and try to say something about her math­em­at­ic­al con­tri­bu­tion. This is enorm­ous and I have de­cided to con­cen­trate on one as­pect of it which I know the best, Dowker spaces.

The story of Dowker spaces starts with J. Dieud­onné, who in his 1944 art­icle [e1] con­sidered suf­fi­cient con­di­tions for a Haus­dorff to­po­lo­gic­al space $X$ to sat­is­fy that for every pair $(h, g)$ of real func­tions on $X$ with $h < g$, if $h$ is up­per semi­con­tinu­ous and $g$ is lower semi­con­tinu­ous, then there is a con­tinu­ous $f : X \rightarrow \mathbb{R}$ such that $h < f < g$. C. H. Dowker in his 1951 art­icle [e2] showed that this prop­erty is equi­val­ent to the product $X \times [0, 1]$ be­ing nor­mal (an­swer­ing a con­jec­ture of S. Ei­len­berg) and left open the pos­sib­il­ity that this was simply equi­val­ent to X be­ing nor­mal. A pos­sible counter­example, so a nor­mal space $X$ for which $X \times [0, 1]$ is not nor­mal, be­came known as a Dowker space. In 1955, Mary El­len Rud­in [1] showed that one can ob­tain a Dowker space from a Suslin line. At the time, the con­sist­ency of the ex­ist­ence or nonex­ist­ence of a Suslin line, of course, had not yet been known. M. E. Rud­in re­turned to the prob­lem in 1970 [2] when she gave an in­geni­ous ZFC con­struc­tion of a Dowker space as a sub­space of the product $\prod_{n\geq 1} (\omega_n + 1)$ in the box to­po­logy. This space has car­din­al­ity $2^{\mathfrak{N}_{\omega}}$. This con­struc­tion of Mary El­len in­spired a large amount of re­search, out of which I men­tion the two dir­ec­tions that are most in­ter­est­ing to me.

The first one is Z. Ba­logh’s con­struc­tion [e3] of a “small Dowker space,” which was in this con­text taken to mean a Dowker space of car­din­al­ity $2^{\mathfrak{N}_0}$ (which, of course, in some uni­verses of set the­ory is as large as $2^{\mathfrak{N}_\omega}$ but in oth­ers it is not). Ba­logh’s meth­od is a very in­tric­ate use of cer­tain se­quences of func­tions and ele­ment­ary sub­mod­els, and read­ing it even today, one feels that its full lim­its have not yet been reached. The oth­er dir­ec­tion was taken by M. Ko­j­man and S. She­lah in [e4], where they used pcf the­ory to find a sub­space of M.  E. Rud­in’s space which is still Dowker, but has size $\mathfrak{N}_{\omega+1}$. This con­struc­tion rounds up the long list of set-the­or­et­ic tech­niques (for­cing, large car­din­als, pcf, …) that were used in Mary El­len’s work and the work she in­spired, car­ry­ing the field of gen­er­al to­po­logy to what has since be­come known as set-the­or­et­ic to­po­logy. The in­ter­ac­tion between set the­ory and set-the­or­et­ic to­po­logy has been most fruit­ful and has car­ried both sub­jects for­ward. Mary El­len was the moth­er of it all. Let us hope that the un­ruly chil­dren that we are, we shall be able to carry for­ward not only Mary El­len’s math­em­at­ic­al leg­acy but also the leg­acy of the spir­it of col­lab­or­a­tion, in­spir­a­tion, mu­tu­al re­spect and joy for all which she was able to cre­ate and live up to.

### Works

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

[2] 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