three of its subcategories, and a handful of functors; but to consider them
as instances of more general notions gives us a platform to stand on that is
often welcome.
In 1952 Shirota [l] established the first deep theorem in uniform spaces,
depending on theorems of Stone [l] and Ulam [l]. Except for reservations
involving the axioms of set theory, the theorem is that every topological
space admitting a complete uniformity is a closed subspace of a product of
real lines. A more influential step was taken in 1952 by Efremovic [l] in
creating proximity spaces. This initiated numerous significant Soviet con-
tributions to uniform and proximity geometry (which are different but coin-
cide in the all-important metric case), central among which is Smirnov's
creation of uniform dimension theory (1956; Smirnov [4]). The methods of
dimension theory for uniform and uniformizable spaces are of course mainly
taken over from the classical dimension theory epitomized in the 1941 book
of Hurewicz and Wallman [HW]. Classical methods were pushed a long way
in our direction (1942-1955) by at least two authors not interested in uniform
spaces: Lefschetz [L], Dowker [l; 2; 4; 5]. These methods—infinite coverings,
sequential constructions—were brought into uniform spaces mainly by Isbell
[1; 2; 3; 4] (from 1955).
Other developments in our subject in the 1950's do not really fall into a
coherent pattern. What has been described above corresponds to Chapters I,
II, IV and V of the book. Chapter III treats function spaces. The material is
largely classical, with additions on injectivity and functorial questions from
Isbell [5], and some new results of the same sort. The main results of Chap-
ters VI (compactifications) and VIII (topological dimension theory) are no
more recent than 1952 (the theorem Ind = dim of Katetov [2]).
The subject in Chapters VII and VIII is special features of fine spaces,
i.e., spaces having the finest uniformity compatible with the topology. Chap-
ter VII is as systematic a treatment of this topic as our present ignorance
permits. Central results are Shirota's theorem (already mentioned) and
Glicksberg's [2] 1959 theorem which determines in almost satisfactory terms
when a product of fine spaces is fine. There is a connecting thread, a functor
invented by Ginsburg—Isbell [l] to clarify Shirota's theorem, which serves
at least to make the material look more like uniform geometry rather than
plain topology. There are several new results in the chapter (VII. 1-2, 23,
25, 27-29, 32-35, 39); and a hitherto unpublished result of A. M. Gleason
appears here for the first time. Gleason's theorem (VII. 19) extends previous
results due mainly to Marczewski [1; 2] and Bokstein [l]. He communicated
it to me after I had completed a draft of this book including the Marczewski
and Bokstein theorems; I am grateful for his permission to use it in place
of them.
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