There’s a lot to like about plain old, everyday space. No matter where you are, there’s always a way to get to where you need to go, and you can always figure out how long any route from here to there might be. Calculus works in it, and that means we can solve a lot of problems about how stuff moves through it. No matter what scale you exist at, you never lack for neighbours. Yes, space is pretty great, but what, mathematically, is it about normal space that lets it be that way, and what can you take away from normal space and still have it mainly do the things we want it to?
Many branches of mathematics investigate the question of the properties of regular space, and the theoretical properties of spaces, with slightly different ground rules than the ones we are used to, but one of the most exciting of the last century and a half has been that of set topology. Here, the basic necessities for space-as-we-know-it have been minutely investigated and then, one by one, removed, to see what happens, like a multidimensional version of Kerplunk.
To be a set topologist, dreaming up new spaces with exotic properties and exhilaratingly odd behaviours, requires outside-the-box thinking of an almost unworldly degree and a genius for thinking across different mathematical landscapes as you hunt for exotic widgets and tools that combine to create a particular world with particular aspects heretofore deemed impossible. Because their work is wrapped in an almost impenetrable veil of mathematical language, the great set topologists are all but unknown to most of the world, but within the grand temple of mathematics, they are regarded with the same hushed reverence we reserve for a Shakespeare or a Sam Kerr. Chief among these ‘How does a brain like that even happen?’ legends is Mary Ellen Rudin (1924–2013), a topologist who described herself as part of the ‘housewife generation’ of mathematics but whom other mathematicians described as an individual of breath-taking insight and originality, who defined for generations to come the course of set topology.
Photo: Mathematical Association of America
Mary Ellen Estill was born on 7 December 1924 in the small town of Hillsboro, Texas. Her mother was a high school English teacher and her father a civil engineer whose work for the Texas Highway Department kept the family constantly on the move until the Great Depression trapped them in the rural locale of Leakey, Texas, which boasts a current population of 446 individuals. It was the sort of town that nobody ever much visited and where nothing much ever happened; where children still went to school on horseback, and spent their afternoons in imaginative play and their nights gathered around the family radio set.
Mary’s childhood was a slice of Depression-era Americana, and she expected nothing more from her future than a modest teaching position and a traditional marriage. One of five graduating students in her high school class, when she went on to college at the University of Texas in Austin, her plan, encouraged by her parents, was to take a general class load that would prepare her in a non-specific way for whatever life might offer, but fate had something else in store.
In those days, to register required physically going to a central location where each department had a table set up where you could ask a faculty member what the different classes were and sign up for the ones you wanted. When Mary arrived on registration day, it was not with any particular agenda or wish list in mind, and so she was ushered to the table with the shortest queue: that of the mathematics department. Here, there just happened to be seated Robert L. Moore, one of the most eccentric and controversial figures in American mathematics. He talked with Mary about maths, about the basic elements of proof, like ‘if-then’ statements, and determined, based on that small interview, that Mary had a logically prone mind that needed development. Moore took her under his wing, and until her senior year every maths course she took was taught by Moore.
This was both a good and bad thing. Moore absolutely discouraged his students from reading maths papers put out by other mathematicians, and instead wanted them to cultivate their ability to solve problems from the ground up, using only starting definitions and their own innate sense for how mathematics works. The result was a generation of ‘Moore mathematicians’ who emerged from college with an incredible and individualistic ability to grapple with complex mathematical problems confidently, but who also possessed huge gaps in their knowledge base and no familiarity with the current mathematical literature.
At the end of her three years as an undergraduate, Estill was invited by Moore to continue at the University of Texas for graduate school, with himself as her advisor. Professionally, this was perhaps a disservice to Estill, who as a graduate student needed to be exposed to other styles and mathematical approaches, and to broaden the base of her mathematical knowledge, rather than be led deeper into Moore’s idiosyncratic system with its non-conventional notations.
Though his push to have Estill stay under his supervision at the University of Texas was professionally a disservice, it was, from a human point of view, understandable. As Estill later reflected, she represented for him an opportunity that he almost never had, a brilliant and original mind as yet completely untouched by standard approaches to advanced mathematics; a mind that could be completely moulded along the lines of his principles and then unleashed to see how far it might go.
Photo: American Mathematical Society
Whether he was honestly convinced she could only flourish under his guidance, or whether he was just reluctant to lose his greatest pupil, Moore ensured she stayed well within his orbit for as long as reasonably possible. When she wrote her graduate thesis in 1949, it featured the creation of a brilliant counterexample to one of Moore’s own axioms, and was couched in his own virtually impenetrable mathematical symbols. It showed the great promise of her mind, but also was limited in its impact, thanks to the language in which it was couched.
When it came time for Estill to leave the University of Texas, she was told by Moore that he had already arranged for her to attend Duke the next year to teach at its women’s college. Here she met her future husband, Walter Rudin, an Austrian emigrant who had fled the Nazis in the Second World War, and who was also a promising young mathematician. When they married in 1953 she took it as a matter of course that she would follow him to the University of Rochester, where he worked as a professor, and become what she called a ‘housewife mathematician’ – a woman who did the work of raising children and having a family, but who in her spare moments dedicated herself to mathematics and part-time teaching.
Mary Ellen Rudin had her first child in 1954, and her second in 1955, aided by an indefatigable nanny, Lila Hilgendorf, whose help gave Rudin the space she needed to pursue her intellectual goals – which included in 1955 the feat for which she is most known: the creation of a Dowker space.
Dowker spaces feature two properties that don’t play well together: normality and a lack of countable paracompactness. Usually, if a space is normal (if, given two points in that set, you can always find little neighbourhoods of those points that don’t intersect with each other), it is countably paracompact (given a bunch of sets that cover up the whole space, you can always create a finer cover of that space that is locally finite). Compactness and normality are two of the biggest supports for making the space we know and love work the way it does, and they tend to be fellow travellers. So, a space that is normal but not countably paracompact would be a rare creature indeed, and in 1951 Clifford Dowker theorised one could not be made, a theory which held until Rudin strolled on up and made one.
Her original Dowker space, from 1955, was based on a mathematical object which, at the time, mathematicians were not sure of the existence of: Suslin lines. This creation, which harnessed an object which might or might not exist to construct an object that had been declared impossible, did not catch on, because it existed outside the realm of the Zermelo-Fraenkel set theory (or ZFC), which forms the basis of the axiomatic system in which discussions about sets and their properties are generally couched.
Later, in 1970, Rudin would strike again with Dowker spaces, constructing a massive Dowker space within the standard confines of ZFC. Here, she made a Dowker space from a massive subspace of a particular product space under the box topology (a space’s topology being defined as the way that we define ‘neighbourhoods’ around the points in that space, which itself affects how distance, separation and fineness are defined in that space), and thereby made the first ZFC Dowker space in two decades of trying, setting a new course for the bleeding edge of set topology.
The Rudins had moved from the University of Rochester to the University of Wisconsin in 1959, where Mary Rudin continued as a part-time teacher and researcher, but after her 1970 feat of finding a ZFC Dowker space, the university could hardly stick by the letter of old anti-nepotism laws any longer, and offered her a full professor position. Walter wanted her to take it, so that she would have a means of financial support if he were to suddenly pass away, but Mary herself was not so sure. As it stood, her work consisted precisely of those things she wanted to do – researching absurdly difficult mathematics problems and teaching now and then – without any of the things that came with being a full professor that she decidedly did not want to do – committee work, departmental politics, and the mad scramble for prestige and power. Her ideal, expressed in later interviews, was to sit in the middle of the living room in her and Walter’s Frank Lloyd Wright-designed home, with children rambling about everywhere while she sat on the couch and drew her way through intriguing mathematical problems, at the centre of the family, and with plenty of time to work on maths and just maths.
She ultimately took the position, and continued to live as she had, releasing blockbuster mathematical constructions at a regular pace through the next few decades that left her colleagues dumbstruck with the originality of her mathematical inventions. Peter Nyikos, in his memorial remembrance of Rudin’s work for the American Mathematical Society, expressed this sense of wonder:
Her paper on her screenable Dowker space solved a 1955 problem of Nagami whether every normal, screenable space is paracompact. 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 ♢++ 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 ♢++ 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?’
Mary Ellen Rudin’s contributions to mathematics were many, including the first non-trivial proofs about box topologies, the creation of an example that demonstrated one unproven consequence of Gödel’s Axiom of Constructibility, and a proof of Nikiel’s 1986 conjecture that every monotonically normal compact space is the continuous image of a linearly ordered compact space made in 1999, when she was 75 years old and had been retired for ten years. She was the reigning wizard of mid- to late twentieth-century set topology, and a respected mentor who, perhaps because of her own experience as a student having one person impose their mathematical personality on her path, was known for giving students space to work on their problems in addition to a thorough grounding in fundamental principles. She was a phenomenon, a mathematical mind unexplainable in mere terms of hard work and diligence, one of those genius presences placed on Earth from time to time seemingly to remind ourselves just what a human brain can do, and how much we have left to learn.
Mary Ellen Rudin died at her home on 18 March 2013
FURTHER READING:
The American Mathematical Society memoirs of Mary Ellen Rudin feature a number of mini-memorials about her work and personality that palpably demonstrate how entirely ‘other’ her mind appeared to her colleagues. There is also a more through-composed biography and appreciation of Rudin to be found in Claudia Henrion’s Women in Mathematics: The Addition of Difference (1997), while Rudin’s most famous work, her ZFC Dowker construction, can be found online.
And if you want to read more stories about awesome women mathematicians, my book A History of Women in Mathematics is now out, with over a hundred figures, from Pen and Sword Books, so dig in!
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