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 to satisfy that for every pair of real
functions on with , if is upper semicontinuous and is lower
semicontinuous, then there is a continuous
such that .
C. H. Dowker
in his 1951 article
[e2]
showed that this property
is equivalent to the product 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 for which 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
in the box topology. This space
has cardinality . 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.