Celebratio Mathematica

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.