PREFACE

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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