Celebratio Mathematica

There is an ex­cel­lent bio­graph­ic­al in­ter­view of Mary El­len pub­lished in [e2]. Also, there is an en­tire volume [e3] de­voted to art­icles de­scrib­ing her math­em­at­ics. All quo­ta­tions be­low are taken from [e3]. Mary El­len Rud­in was one of the lead­ing to­po­lo­gists of our time. Be­sides solv­ing a num­ber of well-known out­stand­ing open prob­lems, she was a pi­on­eer in the use of set-the­or­et­ic tools. She was one of the first to ap­ply the in­de­pend­ence meth­ods of Co­hen and oth­ers to pro­duce in­de­pend­ence res­ults in to­po­logy. She did not do for­cing ar­gu­ments her­self, but many of her pa­pers make use of set-the­or­et­ic state­ments, such as Mar­tin’s Ax­iom, Suslin’s Hy­po­thes­is, and \( \diamond \), that oth­er re­search­ers have shown to be true in some mod­els of set the­ory and not in oth­ers.

In her thes­is [1] she gave an ex­ample of a non­sep­ar­able Moore space that sat­is­fies the count­able chain con­di­tion. She pub­lished the res­ults of her thes­is in three pa­pers in the Duke Math Journ­al [2], [3], [4]. To quote Steve Wat­son, “This cycle rep­res­ents one of the greatest ac­com­plish­ments in set-the­or­et­ic to­po­logy. However the math­em­at­ics in these pa­pers is of such depth that, even forty years later, they re­main im­pen­et­rable to all but the most di­li­gent and pa­tient of read­ers.”

In [5] she used a Suslin tree to con­struct a Dowker space. The ex­ist­ence of a Suslin tree is con­sist­ent with ZFC but not prov­able from ZFC. In [6] [8] she con­struc­ted a Dowker space just in ZFC. This space has car­din­al­ity \( (\mathfrak{N}_\omega )^{\mathfrak{N}_0} \), where­as the Suslin tree yields a Dowker space of size only \( \mathfrak{N}_1 \). Also, in 1976 [13] she showed that CH yields a Dowker space of size \( \mathfrak{N}_1 \). It is still an open ques­tion wheth­er one can prove in ZFC that there is a Dowker space of size \( \mathfrak{N}_1 \). There is one of size \( \mathfrak{N}_{\omega+1} \) by Ko­j­man and She­lah [e4]. Her work on Dowker spaces led to an in­vited ad­dress at the In­ter­na­tion­al Con­gress of Math­em­aticians in 1974. It also led to her in­terest in the box to­po­logy, be­cause her Dowker space of size \( (\mathfrak{N}_{\omega})^{\mathfrak{N}_0} \) is a spe­cial kind of box product.

Mary El­len is fam­ous for her work on box products. If \( X_n \) (for \( n \in \omega \)) are to­po­lo­gic­al spaces, then \( \square_n X_n \) de­notes the product of the spaces \( \prod_n X_n \) us­ing the box to­po­logy; a base for this to­po­logy is giv­en by all sets \( \prod_n U_n \), where each \( U_n \) is open in \( X_n \). The more well-known Ty­chon­ov to­po­logy, used to prove the Ty­chon­ov The­or­em (1935), re­quires that \( U_n = X_n \) for all but fi­nitely many \( n \). Mary El­len was the first per­son to prove any­thing non­trivi­al about box products. In [10] she showed that, as­sum­ing \( \mathrm{CH} \), \( \square_n X_n \) is nor­mal, and in fact para­com­pact, whenev­er all the \( X_n \) are com­pact met­ric spaces. Also, her 1974 pa­per [11] shows that \( \square_n X_n \) is para­com­pact whenev­er all the \( X_n \) are suc­cessor or­din­als; this eas­ily gen­er­al­izes to the case where the \( X_n \) are com­pact scattered spaces (see [e1]). It was later shown by Eric van Douwen, in ZFC, that there are al­ways com­pact Haus­dorff \( X_n \) for which \( \square_n X_n \) is not nor­mal. The ques­tion of which of Mary El­len’s pos­it­ive res­ults can be proved in \( \mathrm{ZFC} \) re­mains an out­stand­ing un­solved prob­lem in to­po­logy. Even the case where all \( X_n = \omega + 1 \) re­mains open, al­though this case (or when all \( X_n \) are com­pact met­ric) does fol­low from Mar­tin’s Ax­iom.

Mary El­len is also fam­ous for her work on \( \beta \mathbb{N} \), the space of ul­tra­fil­ters on the nat­ur­al num­bers, start­ing in 1966. She was coin­vent­or of two well-known par­tial or­ders on this space: the Rud­in–Keisler or­der and the Rud­in–Fro­lik or­der. The ba­sic prop­er­ties of these are giv­en in her 1971 pa­per [7], al­though their his­tor­ic roots go back some­what earli­er.

The Rud­in–Fro­lik or­der led to the first proof in ZFC that the space of non­prin­cip­al ul­tra­fil­ters, \( \mathbb{N}^\ast = \beta\mathbb{N} \backslash \mathbb{N} \), is not ho­mo­gen­eous. Un­der CH this was already known by a res­ult of Wal­ter Rud­in (1956), who proved from CH that there is a P-point \( \mathcal{U} \in \mathbb{N}^\ast \) (that is, \( \mathcal{U} \) is in the in­teri­or of every \( G_\delta \) set con­tain­ing \( \mathcal{U} \)). Non­homo­gen­eity fol­lows be­cause every in­fin­ite com­pactum also has a non-P-point \( \mathcal{V} \), and no homeo­morph­ism of \( \mathbb{N}^\ast \) can move this \( \mathcal{V} \) to Wal­ter’s \( \mathcal{U} \). It was shown much later (She­lah, in the 1970s) that one can­not prove in ZFC that these P-points ex­ist.

For the Rud­in–Fro­lik or­der, for \( \mathcal{U},\mathcal{V}\in \mathbb{N}^\ast \), say that \( U < _{\mathrm{RF}}\mathcal{V} \) iff \( \mathcal{V} \) is a \( \mathcal{U} \)-lim­it of some dis­crete \( \omega \)-se­quence of points in \( \mathbb{N}^\ast \). One can show that this is a par­tial or­der; also, if \( \mathcal{U} < _{\mathrm{RF}}\mathcal{V} \), then no homeo­morph­ism of \( \mathbb{N}^\ast \) can move \( \mathcal{V} \) to \( \mathcal{U} \), prov­ing non­homo­gen­eity.

The Rud­in–Keisler or­der comes nat­ur­ally out of the no­tion of in­duced meas­ure. If \( \mathcal{U}, \mathcal{V} \) are ul­tra­fil­ters on any set \( I \), we say that \( \mathcal{U} \leq_{\mathrm{RK}}\mathcal{V} \) iff there is a map \( f : I \rightarrow I \) that in­duces the meas­ure \( \mathcal{U} \) from \( \mathcal{V} \); that is, \[ \mathcal{U} = \{X \subseteq I : f^{-1} (X) \in \mathcal{V} \} .\] When \( U, \mathcal{V} \in \mathbb{N}^\ast \), \( \mathcal{U} < _{\mathrm{RF}} \mathcal{V} \) im­plies that \( U \leq_{\mathrm{RK}} \mathcal{V} \) and \( \mathcal{V} \nleq_{\mathrm{RK}}\mathcal{U} \). This or­der is of im­port­ance in mod­el the­ory, in the the­ory of ul­traproducts, since \( \mathcal{U} \leq_{\mathrm{RK}} \mathcal{V} \) yields a nat­ur­al ele­ment­ary em­bed­ding from \( \prod_i \mathfrak{A}_i /\mathcal{U} \) in­to \( \prod_i \mathfrak{A}_i /\mathcal{V} \). It is also im­port­ant in the com­bin­at­or­ics of ul­tra­fil­ters, since the Rud­in–Keisler min­im­al ele­ments of \( \mathbb{N}^\ast \) are pre­cisely the Ram­sey ul­tra­fil­ters; such min­im­al ele­ments ex­ist un­der CH but not in ZFC. It is true, but not com­pletely trivi­al, that neither of these two or­ders is a total or­der; that is, in ZFC there are \( \mathcal{U}, \mathcal{V} \in \mathbb{N}^\ast \) such that \( \mathcal{U} \nleq_{\mathrm{RK}} \mathcal{V} \) and \( \mathcal{V} \nleq_{\mathrm{RK}} \mathcal{U} \) (and hence also \( \mathcal{U} \nless_{\mathrm{RF}} \mathcal{V} \) and \( \mathcal{V} \nless_{\mathrm{RF}} \mathcal{U} \)). The pa­per [14] (1979) of Mary El­len and Sahar­on She­lah gives the strongest pos­sible res­ult: for each in­fin­ite \( \kappa \), there is a fam­ily of \( 2^{2^\kappa} \) ul­tra­fil­ters on \( \kappa \) that are pair­wise in­com­par­able in the Rud­in–Keisler or­der­ing. Mary El­len worked ex­tens­ively on the ques­tion of \( S \)- and \( L \)-spaces. She pro­duced the first \( S \)-space (a hered­it­ar­ily sep­ar­able space that is not hered­it­ar­ily Lindelöf) as­sum­ing the ex­ist­ence of a Suslin tree in 1972 [9]. Her Dowker space con­struc­ted from CH (see above) is also an \( S \)-space. Her 1975 mono­graph [12] de­votes an en­tire chapter to \( S \)- and \( L \)-spaces. To quote Stevo To­dorčević, “The terms ‘\( S \)-space’ and ‘\( L \)-space,’ which are pre­dom­in­ant in most of the lit­er­at­ure on this sub­ject, are also first found in these lec­tures. This shows a great in­flu­ence not only of [12] but also of M. E. Rud­in’s per­son­al­ity on the gen­er­a­tion of math­em­aticians work­ing in this area, since it is rather un­usu­al in math­em­at­ics to talk about cer­tain state­ments in terms of their counter­examples.”

In 1999, al­most a dec­ade after her re­tire­ment, Mary El­len settled a long-stand­ing con­jec­ture in set-the­or­et­ic to­po­logy by show­ing that every mono­ton­ic­ally nor­mal com­pact space is the con­tinu­ous im­age of a lin­early ordered com­pact space. This pa­per [15] was the fi­nal one in a series of five which gradu­ally settled more and more spe­cial cases of the fi­nal res­ult. The con­struc­tion of the lin­early ordered com­pact space is ex­traordin­ar­ily com­plex, and to this date there is no sim­pler proof known.

Mary El­len was by con­sensus a dom­in­ant fig­ure in gen­er­al to­po­logy. Her res­ults are dif­fi­cult, deep, ori­gin­al, and im­port­ant. The con­nec­tions she found between to­po­logy and lo­gic at­trac­ted many set-the­or­ists and lo­gi­cians to to­po­logy. The best gen­er­al to­po­lo­gists and set-the­or­ists in the world passed reg­u­larly through Madis­on to vis­it her and work with her and her stu­dents and col­leagues.

She had eight­een PhD stu­dents, many of whom went on to have ster­ling ca­reers of their own. To quote one of them, Mi­chael Star­bird, “From the per­spect­ive of a gradu­ate stu­dent and col­lab­or­at­or, her most re­mark­able fea­ture is the flood of ideas that is con­stantly burst­ing from her…. It is easy to use the Mary El­len Rud­in mod­el to be­come a great ad­visor. The first step is to have an end­less num­ber of great ideas. Then merely give them totally gen­er­ously to your stu­dents to de­vel­op and learn from. It is really quite simple. For Mary El­len Rud­in.”