#### by Peter Nyikos

I had the honor of delivering a short eulogy for Mary Ellen at this
year’s Spring Topology and Dynamics Conferences (STDC) in Connecticut.
It said: “Mary Ellen was a great mathematician. But she was much more
than that: she was what
Prabir Roy
called a guru — someone you could
turn to for advice and comfort on all kinds of matters… To those of
us who knew her well, she was simply ‘Mary Ellen.’ Whenever
set-theoretic topologists got together for a chat, and someone said
“Mary Ellen,” 99 times out of a hundred, everyone would know who was
being talked about.” I briefly listed some of her main accomplishments, including of course the writing of “Mary Ellen’s booklet”
[1].
another expression that usually gets instant recognition.
Among her research accomplishments is a beautiful generalization of
the Hahn–Mazurkiewicz theorem. It is an immediate corollary of her
solution to Nikiel’s Conjecture
[3]
and of a 1988 theorem of
Nikiel
[e3].
The Hahn–Mazurkiewicz theorem states that, if a metrizable
space is a locally connected continuum (compact, connected space),
then it is a continuous image of __\( [0, 1] \)__. (The converse is elementary.)
These spaces are called “Peano continua” in recognition of Peano’s
space-filling curve. Back in 1966
[e2],
Sibe Mardeši
wrote: “It is
natural to ask for a nonmetric analogue of this theorem….Recently
the interest in this and related problems has been revived….” Mary
Ellen’s solution to Nikiel’s Conjecture allows one to substitute
“monotonically normal” for “metrizable” in the Hahn–Mazurkiewicz
theorem. And the proof that every metric space is monotonically normal
is essentially identical to the usual proof that every metric space is
normal. The search for a natural extension of the Hahn–Mazurkiewicz
theorem to compact connected, linearly ordered spaces was akin to the
search for a metrization theorem generalizing the Urysohn
metrization theorem, and it lasted even longer. (The
Bing–Nagata–Smirnov theorem was hailed as the solution to this
metrization problem after a search of over three decades.)

I closed my eulogy by expressing the hope that there would be some
publications in remembrance of Mary Ellen that would do justice to her
greatness, and I am very happy to be able to contribute both to the
special issue of *Topology and its Applications* dedicated to Mary Ellen
and to this remembrance. I got a unique taste of Mary Ellen’s
graciousness and hospitality in early 1974, when I was a postdoctoral
student at the University of Chicago. She invited me up to Madison,
where I arrived with a bad cold (I naïvely decided not to postpone the
visit, which had already been delayed a number of times), but although
it was obvious to everyone, she never mentioned it once and had me
stay overnight at her house, where I met her two sons and played board
games with them. The same evening she introduced me to the axioms __\( \diamond \)__
and __\( \clubsuit \)__ and to Ostaszewski’s __\( S \)__-space, all of which were totally new to
me at the time. The next two years I saw her at the two STDC
conferences, where I became impressed first by her lecturing style and
then by the high regard in which she was held. There was a panel
discussion in Memphis about the future of point-set topology, and the
panel included Mary Ellen and other leading figures such as
R. H.Bing,
R. D. Anderson,
A. H. Stone,
and
E. Michael.

Her paper on her screenable Dowker space
[2]
solved a 1955 problem
of
Nagami
whether every normal, screenable space is paracompact
[e1].
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 __\( \diamond^{++} \)__ 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 __\( \diamond^{++} \)__ 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?”

There were some problems in set-theoretic topology which Mary Ellen
could not solve but on which she did obtain large “consolation
prizes.” One such prize was her screenable Dowker space, an offshoot
of her unsuccessful attempts to solve a problem for which we still
have no consistency results: is there a normal space with a
__\( \sigma \)__-disjoint base that is not paracompact?
Zoltán Balogh
later
[e4]
came up with one of his “greatest hits”: a ZFC example of a
screenable Dowker space. Mary Ellen’s consistent example is, however,
the only one known to be collectionwise normal. A recurring theme in
Mary Ellen’s research, up to the very end of her life, were two
further problems about Dowker spaces: the problem of whether there is
a normal, linearly Lindelöf space that is not Lindelöf and the problem
of whether there is a normal space with a __\( \sigma \)__-disjoint base that is not
paracompact.