#### by Kenneth Kunen and Arnold Miller

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.”