#### by Mirna Džamonja

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.