Memorial from Volume 6, #1, of TopCom
"I see little use for the elaborations of axiomatic topology. I had a rather severe case of the disease as a young man, but I'm happy to say it has been almost totally cured. Right now, I don't care a bit whether every beta-capsule of type delta is also a T-spot of the second kind."Evidently he had identified a sickness which afflicts much of point-set topology, and he wanted no part of it.
I do not believe that in later life Ed was seriously embarrassed by the misstatements in his early topological work. He simply went on to bigger and better things. We can never know what topological riches he would have given us, had he not quit the fraternity. Rather than mourn imagined losses, let us here take pleasure from his contributions. For reasons of indolence, inadequate memory and constraints of time and space, I will make extended or substantive comments concerning only his two best-known topological papers, [29] and [34], touching briefly on the others. Where appropriate I will borrow freely (with credit) from Hewitt's lively prose and terminology, while profiting nevertheless from the benefits of a consistent, coherent, modern vocabulary (which differs at points from his).
If Part I of [29] might be labeled pedestrian by today's standards, the same criticism cannot be leveled at Parts II and III (for which Part I is essential preparation), which are novel, imaginative, and fertile. Introducing a new concept, Hewitt calls a space X resolvable if it admits complementary dense subsets. The reader of this review might enjoy recapturing his result that every locally compact space which is dense-in-itself (that means: without isolated points; hereafter abbreviated d-i-i), and also every d-i-i metric space, is resolvable. More interesting is this result, based again of course on a "proof [which] reposes upon a variant form of Zorn's theorem", which shows that irresolvable topologies exist in such profusion that (for example) every d-i-i Tychonoff topology on any set expands to an irresolvable d-i-i Tychonoff topology on the same set: Let (X, T) be a regular [resp., completely regular] d-i-i Hausdorff space and let U be a regular [resp., completely regular] topology on X maximal with respect to the properties U Ê T and (X, U) is d-i-i; then (X, U) is irresolvable. The necessity in this context of an appeal to transfinite methods was demonstrated subsequently by El'kin [21], who showed that a space (X, T) is irresolvable if and only if T contains a base for an ultrafilter on X. Thus the problem posed in [11,8.8] - "[w]ith or without the Axiom of Choice, give a concrete example of an irresolvable Hausdorff space without isolated points (the open sets being identified explicitly in concrete form)" -could reasonably be dismissed by Malykhin [53] as unrealistic or "hardly possible". In this connection, however, see the later work [2,2.3], where a countable, dense irresolvable subspace of {0,1}c is displayed-using heavily, of course, the "C" of "ZFC".
Following the publication of [29], Hewitt never returned to the concept of resolvability. Since a few years would pass before he introduced the concept of pseudocompactness [34], the following question was never posed by Hewitt himself. In every real sense it belongs to him, however, since it combines naturally two useful, fruitful concepts of his invention.
Following modern use we drop the hyphen used by Hewitt [34] in defining what he called a pseudo-compact space. We say simply that X is pseudocompact if every real-valued continuous function on X is bounded.
Question [11]. Is every d-i-i pseudocompact space resolvable?
Two significant contributions to this question were given, albeit peripherally since their principal interest lay elsewhere, by Kunen, Szyma\'nski and Tall [48]. They showed under V = L that every Baire space (in particular, every Tychonoff pseudocompact space) without isolated points is resolvable. (See also [49] for an elaboration of their proof, and see [2] for a proof in ZFC that the irresolvability of every Baire space is equivalent to the condition that every submaximal space is s-discrete.) They showed also [48] that if the existence of a measurable cardinal is consistent with ZFC, then so is the existence of a (d-i-i, zero-dimensional) irresolvable Baire space X; the spaces X they find however are not pseudocompact, so the question above remains open in ZFC. In any case there are difficulties in constructing an irresolvable pseudocompact Tychonoff space: As noted above there is an intimate relation between irresolvability and maximal topologies, and according to [11,7.7] no maximal d-i-i Tychonoff space can be pseudocompact; further, every d-i-i countably compact Tychonoff space is not only resolvable [13,5.2] but even w-resolvable [11,6.9] in the sense of the following definition.
Among his other contributions in [29], Hewitt coined the expression extremally disconnected to describe those topological spaces in which disjoint open sets have disjoint closures. (The concept had been introduced in 1937 by M. H. Stone [69], who showed that the Tychonoff spaces X with that property are precisely those for which the ring C(X) is conditionally complete in the sense that each of its subsets which is bounded above has a least upper bound. The same result was given independently by Nakano [55].) Hewitt [29] provided several additional characterizations, and he showed that every maximal Tychonoff space is extremally disconnected [29]. Once again, two disparate properties brought to our attention by Hewitt were shown unexpectedly to be related: Malykhin [52], working in the axiom system ZFC augmented by the combinatorial principle P(c), produced an example answering affirmatively a well-known, difficult question of Arhangel'ski [4]: Does there exist a non-discrete, extremally disconnected topological group? Malykhin's example is the countable Boolean group B := Åw {0,1} with a topology T which is, in fact, unlike earlier examples given by Sirota [67] and Louveau [50] using stronger supplementary axioms, maximal not only among Hausdorff d-i-i group topologies on B but even among regular Hausdorff d-i-i topologies on B. Thus, according to two of the results of Hewitt [29] cited above, the group (B,T) is both irresolvable and extremally disconnected. Complementing Malykhin's result, it was later shown [14] in ZFC that every Abelian group not containing algebraically a copy of the group B is resolvable in every non-discrete group topology. This result has been strengthened by Zelenyuk [72]: every such Abelian group is absolutely resolvable in the sense that it contains a pair of disjoint subsets each of which is dense in every non-discrete group topology. Earlier Malykhin and Protasov [54] had given a result of much the same flavor: Every (not necessarily Abelian) infinite group G admits a family A of |G|-many pairwise disjoint sets each of which is dense in every totally bounded (i.e., pre-compact) Hausdorff group topology on G. (That theorem is perhaps a trifle less startling than it at first appears, since a group G which admits a Hausdorff totally bounded group topology admits a largest such topology t; thus it is enough to find |G|-many disjoint sets each of which is t-dense. Oddly, Malykhin and Protasov [54] do not need or use that observation. Beginning with an arbitrary well-ordering of the set of finite subsets of G, they define by a pleasing but straightforward inductive argument the elements of a family A as required. A readable version of their proof and of related results appears in [12].)
One must not, of course, credit Hewitt with the many variations on the themes of [29] which were set down by later authors. The remarks above, however, reveal Hewitt the topologist as both imaginative in his own right and provocative of extensive subsequent research; the cliché "seminal" springs to mind as fairly descriptive of his first work [29]. In any case, resolvability has become an honorable subject of research possessed of a substantial momentum. The last chapter lies many years down the road.
How well-behaved can be a space X (with |X| > 1) which is so ill-behaved that every continuous function f:X ® R is constant? Obviously X cannot be a Tychonoff space; Urysohn [70] and Pospísil [60] found Hausdorff spaces X, countably infinite and of arbitrary pre-assigned infinite cardinality respectively, of this type, but their examples are not regular. Responding to a question posed explicitly by Urysohn [70], Hewitt [31] shows that there exist "in great profusion" regular Hausdorff spaces on which each such function is constant, indeed there are such spaces of arbitrary cardinality k ³ w with cf(k) > w. Hewitt finds also a countable connected space Y which is a Urysohn space in the sense that each pair of distinct points admits disjoint closed neighborhoods. I believe that Dieudonné's remark [17] that "[r]elativement aux axiomes de séparation actuellement connus, ces résultats ne peuvent être améliorés" may be freely translated "relative to the separation properties now available in the literature, these results are best possible"; this is in consonance with Hewitt's comment [31,p. 509] that a space with the properties of Y cannot in addition be regular, since as noted by Urysohn [70] every connected regular (Hausdorff) space is uncountable.
"What about the product of countably many separable spaces, for example Rw - is that space separable?",but someone near the middle of the class will remark with satisfaction that
"well, Qw must be dense in Rw".Of course that observation is correct, but since |Qw| = c > w it does not answer the question. Eventually, within the hour or a day or two later, a proof will emerge from the class that indeed Rw does admit a countable dense subset. But nobody of sound mind will believe the instructor who asserts that even Rc is separable. That remarkable statement is, however, a special case of the so-called Hewitt-Marczkewski-Pondiczery theorem (cf. [32], [56], [59]), which in its optimal form, combining aspects of all three treatments, may be stated as follows:
Given an infinite family {Xi: i Î I} of spaces, each admitting a pair of disjoint nonempty open sets, the density character of the space X := Pi Î I Xi is given by the formulaThus indeed, as noted explicitly by Hewitt [32], spaces such as {0,1}c and RR are separable.d(X) = max{ log|I|, sup{d(Xi): i Î I}},where as usual for k ³ w one writes log(k) = min{a: 2a ³ k}.
We need not linger long here over the relative strengths of the three
treatments of the surprising H-M-P result.
It is enough to remark that both Hewitt [32] and the
Pondiczery consortium [59]
showed not only that the product
of £ 2k-many
spaces of density character
£ k
again hasdensity character
£ k,
but also that no product of more than
2k-many non-degenerate
(Hausdorff, say) spaces has that property.
Marczewski [56], though he
touched only the case
k = w,
derived in addition the pretty corollary
(page 142) that no product of separable spaces admits a family of
uncountably many pairwise disjoint nonempty open subsets.
A pleasing, direct and self-contained measure-theoretic proof of that
theorem was given later by Oxtoby
[57], but it is unclear
how to generalize his specialized argument to arbitrary
k ³ w.
In contrast, Marczewski's argument works equally well to show that
in an arbitrary product
Pi Î I
Xi with each
d(Xi) <
His lectures combined a respect for the subject, almost a reverence, with
an occasional arresting bawdy anecdote.
When appropriate while preparing to introduce an important result, he
would regain our wandering attention with the warning
"Now pay attention, kiddies, this is one of the Seven Pillars of
Analysis."
As diligent students we wanted, of course, an accurate and complete list
of these, but this was never forthcoming-nor are any specific theorems
in his books [42], [43] and [44] so designated.
The Stone-Weierstrass theorem, the inequalities of Minkowski
and Hölder and Cauchy-Schwarz-Bunyakowski, the
Riesz representation theorem-these qualified of course. Exhaustive
checks with some of the more senior graduate students of that era who had
studied an earlier course with Hewitt, as well as subsequent conversations
with some who followed later, reveal that the Seven Pillars are at least
twelve in number.
Alas, the official definitive list as sanctioned by Hewitt himself was
never compiled.
In the comments cited below from [10], written in 1976, I
attempted to capture some of the diversity of the developments which
flowed subsequently from Hewitt's insights.
Knowing that Hewitt was a careful thinker who left little to chance,
I figured that his choice of the term Q-space to designate those
spaces he had introduced was driven by some etymological subtlety which
had escaped me.
Accordingly I asked him, several years after [34] was published,
how he had arrived at the name.
He asserted that indeed he had given the matter much thought, and that
when his manuscript was nearing completion he had found himself
dissatisfied with each of the various names he was considering.
Seeking a suitably erudite yet brief and informative term, Hewitt, who was
enjoying a Guggenheim Fellowship at Princeton at the time, finally sought
advice from a colleague in the Classics Department.
He described as best he could in layman's language the mathematical
properties enjoyed by the spaces he wished to name.
After some reflection, his colleague proposed a term with strong Greek
flavor, some six syllables long.
Hewitt expressed his gratitude and returned home to sleep on the matter.
He awoke the next morning, he told me, clear of mind and happily decisive:
"To hell with it. I'll just call 'em Q-spaces."
I suppose that the "Q" was chosen to resonate with the English word
quotient, but that I do not know for sure.
Concerning the notation uX, introduced in [34] to
designate the set of points in bX to which each function in C(X)
extends continuously (so that X is realcompact if and only if
X = uX), I wrote in [10] that
Although characterized later by Hewitt as "lukewarm" [41], Dieudonné's
review [18] of
[34] always
struck me as a model of fairness and good reporting, appreciative and
complimentary but with its enthusiasm necessarily tempered by the
observation that certain of the results claimed were false and others
therefore in doubt.
(Hewitt "proved", unfortunately, that an arbitrary compactification of a
product space
Pi Î I Xi
has the form
Pi Î I Bi
with (for each i Î I) Bi
a compactification of Xi, which led him to
the additional erroneous conclusion
b(Pi
Î I Xi)
= Pi
Î I b(Xi);
instead this latter identity was shown later by
Glicksberg [27], and by
Frolík [23]
when I is finite, to hold if and only if
Pi Î
I Xi is pseudocompact.)
With the advantage of some decades' hindsight I have asked myself,
frequently but always unsuccessfully, how the review [18]
could have been altered to become more fair and balanced.
Despite its several flaws, Hewitt's paper [34] is unquestionably
one of the most far-reaching and fruitful papers ever written in general
topology.
It introduced not only pseudocompact spaces but also (what we now call)
realcompact spaces, the Hewitt realcompactification uX
of an
arbitrary Tychonoff space X, and the realclosed hyper-real fields
C(X)/M (M = Mp,
p Î uX \ X).
According to Schatz [65],
the paper [34] was one of
only 37 papers published in the period 1950-1965, among some 25000 he
surveyed, which had by 1971 been cited by other researchers at least 50
times.
The popular, invaluable book of Gillman and
Jerison [26] describes
elegantly the fruits of many lines
of investigation initiated by Hewitt [34], while continuing to
stimulate non-trivial research even today, some four decades after its own
publication.
I close these comments on Hewitt's paper [34] and its influence by
quoting the phrase used by Hewitt himself [39] in reference to
[26]: it has indeed become
"apotheosized with the passage of
time to the status of a classic."
6 The Stone-Weierstrass Theorem
The theorem known today as the Stone-Weierstrass theorem may be put
in the form
"Every compact space possesses [a certain property] P."
In [33], Hewitt shows that
property P always fails,
if X is not compact; he finds a property Q enjoyed by every
Tychonoff space; and he shows how and why it is that Q reduces
to P in case X is compact.
Quite naturally, Tychonoff spaces X for which
|b(X) \ X| £ 1
command special attention in this context.
These are dubbed "almost compact" spaces by Gillman and
Jerison [26,Problem 6J],
q.v. for a succinct
summary of several characterizations (in terms of the extendability of
real-valued continuous functions) of those spaces as determined by
Hewitt [36] in 1949.
7 Anecdotal Interlude
Mention of the Stone-Weierstrass theorem calls to mind the fact that
as a first-year graduate student in the academic year 1954-55 I had the
privilege of attending Hewitt's introductory graduate Analysis
course.
Hewitt was a dynamic lecturer, always carefully prepared, who convinced us
of the interest and importance of the material at hand by a subtle,
pleasing blend of showmanship with an evident love of the subject and a
rigorous attention to detail.
On one occasion, for example, when a summarizing computation assumed the
unexpected form
|a - b| £ ... < [(9e)/8]
rather than the anticipated, more pleasing
|a - b| £ ... < e,
he took an
additional five minutes of classroom time to revise the various
constituent inequalities before our eyes until the bottom line took on the
desired form.
Was this silly on his part?
Not entirely, I think.
More vividly than any of his occasional direct dicta ever managed, this
exercise conveyed effectively his belief that Mathematics among other
things is an expression of art and aesthetics.
Of course there is no Mathematics whatever without the required
preliminary blood, toil, tears and sweat, but the last 5% or so of a
successful argument or a published paper is in the presentation.
8 Concerning Rings of real-valued continuous functions I; Q-spaces
The chef-d'oeuvre of Edwin Hewitt's work as topologist is the
landmark paper
Rings of real-valued continuous functions I [34].
If he had written no other mathematical papers whatever, he would have
earned a prominent niche in the permanent topological Pantheon on the
basis of this remarkable publication.
It was in order to study this paper that Melvin Henriksen, joined
later by Leonard Gillman and Meyer Jerison, initiated a
topological seminar at Purdue University in 1953.
Recognizing its depth the three men, again at Henriksen's
initiative, began to develop a set of notes clarifying and extending
Hewitt's ideas.
These blossomed eventually into the important book by Gillman and
Jerison [26], Henriksen
having left the cooperation
in deference to his research fellowship at the Institute for Advanced
Study in Princeton during the academic year 1956-57.
In the introduction to their Notes for [26], Gillman
and Jerison write in part
"The groundwork for the theory of rings of continuous functions was laid
in three papers.
The first was Stone's [68],
in which the basic theory of
C* was developed.
...
In the second paper, by Gelfand and
Kolmogoroff [25], it was
shown that some of Stone's
results could be obtained without considering, as Stone had, the
metric structure of the ring C*. This opened the way to a
similar study
of C, which they initiated. Finally Hewitt, in [34], made the
major contributions to the ring C, and set the direction for most of the
subsequent research."
Summarizing and oversimplifying, one may say that Stone, while
giving his construction of the Stone-Cech compactification
bX of an arbitrary Tychonoff space X
[68], realized that
space as the set of zero-set ultrafilters on X, topologized so that an
ultrafilter p (in bX) is in the closure of
a zero-set Z of X if and only if Z Î p;
Gelfand and Kolmogoroff [25]
identified the maximal ideals of C(X) with the points of
bX, via the association
p « Mp :=
{f Î C(X):
p Î
clbXZ(f) };
and Hewitt [34], noticing that
for certain spaces X which admit
unbounded real-valued continuous functions there exist points
p Î bX \ X
to which all such functions extend
continuously, achieved a successful classification of the quotient fields
C(X)/Mp: when p is as above this is naturally isomorphic to
the real
field R, otherwise C(X)/Mp is what Hewitt called
hyper-real-a non-Archimedean, linearly ordered field, properly
containing R as a linearly ordered subfield.
These fields deserved, and received at Hewitt's hand, systematic study as
objects of interest in their own right; as noted by
A. Robinson [62],
[63,2.12, 10.6],
they are readily accessible models of peculiar objects known earlier
through the ultrapower construction of Los.
As is evident from his autobiographical remarks
[41,p. 4],
Hewitt derived much pride and pleasure from revisiting these quotient
fields with Soviet co-authors in the 1981 paper Rings of real-valued
continuous functions. II
[1], some 35 years after the
publication of "I".
Both the review of [1] by
Blass [5] and the
survey article of Henriksen
[28] provide helpful
perspectives on [1], and are
recommended as preliminary
reading; the latter article is an authoritative review of the development
of several aspects of the theory of rings of continuous functions a
quarter-century after the publication of
[26].
"What does it mean to say that a completely regular, Hausdorff space X
is realcompact?
To Edwin Hewitt, who introduced the class of realcompact spaces under the
title Q-spaces, it means that for every maximal ideal M in ...
C(X), either M = {f Î C(X): f(p) = 0}
for some p Î X or the
linearly ordered field C(X)/M is non-Archimedean.
To a point-set topologist, it means that for some cardinal
"I was informed by a letter from Professor Hewitt ...
that he `chose upsilon by some crude association with the word unbounded,
just as Cech probably chose b
because he was thinking of bounded functions'."
In that review I had the temerity
"to respectfully take issue with [Hewitt's] analysis of Cech's
motivation.
...
I suspect that Cech chose the symbol b
for its affinity with
the word `bicompact', or in order to contrast with the notation aX
(used commonly, when X is locally compact but not compact, to denote the
one-point compactification of Alexandroff).
Indeed it seems likely that if Cech had been led to consider with
emphasis the bounded real-valued continuous functions on X, he
might have continued next to the unbounded ones, thus perhaps constructing
on his own an early version of Hewitt's space
uX."
Perhaps today, some 65 years ex post facto, the question of
Cech's motivation resists conclusive disposition.
9 Reactions to Rings of real-valued continuous functions I
In a letter written October 9, 1948 from the University of
Chicago on
stationery marked Universidade do Brasil, Leopoldo Nachbin
requested from Hewitt a reprint of the paper [34], noting that
"[p]art of your results on Q-spaces is identical to part of some
results which I have been working on but not yet published."
Responding three days later, Hewitt indicated a friendly interest in
continuing correspondence and possible collaboration, and he posed one of
"a number of questions which I have not been able to answer, [namely] is
every closed subset of a product of lines a Q-space? My guess is that
the theorem is true, but I cannot prove or disprove it."
That conjecture was established in 1952 by Shirota [66],
the converse implication having already been given by Hewitt
himself [34,Theorem 60].
Following through on Nachbin's remark, Hewitt took the opportunity
in print two years later, while revisiting his Q-spaces with a view to
the study of ring-homomorphisms from rings of the form C(X) to R,
to note that [37,p. 170]
"[t]he same class of spaces has been studied independently, from a
different point of view, and most fruitfully, by Dr. Leopoldo
Nachbin."
In his appreciative study of Nachbin's life and scientific works,
J. Horváth [45,p. 6]
finds this remark
"very fair" on Hewitt's part.
Hewitt in his turn, however, was not pleased with Horváth's
reconstruction of the flow of events surrounding the discovery and
development of these spaces (loc cit.), since that account not only
gives clear priority to Nachbin but also appears to suggest the
possibility that Hewitt had intimations, second- or third-hand, of
Nachbin's work prior to his own publication.
In a letter to Horváth [40] reacting to a
pre-publication typescript of [45], Hewitt politely but firmly
rebutted and rejected this rendering of events, noting that
"[m]y construction was presented to the AMS on 27 April 1946 and
29 December 1946"
and remarking that
"I would be grateful if, in any future version of your biography of
Nachbin, you would set the record straight."
The facts of the matter seem clear to me:
10 Concluding Remarks
In addition to the many far-reaching contributions (partially suggested
above) contained in the remarkable paper [34], Hewitt in his
five-year career as set-theoretic topologist:
This record of achievement is truly outstanding.
The topological community is pleased to recognize and claim Edwin Hewitt
as one of its leaders.
References
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