PREFACE

The subject matter of this book might be labelled fairly accurately Intrin-

sic geometry of uniform spaces. | For an impatient reader, this means elements

(25%), dimension theory (40%), function spaces (12£%), and special topics

in topology.} As the term "geometry" suggests,'we shall not be concerned

with applications to functional analysis and topological algebra. However,

applications to topology and specializations to metric spaces are of central

concern; in fact, these are the two pillars on which the general theory stands.

This dictum brings up a second exclusion: the book is not much concerned

with restatements of the basic definitions or generalizations of the funda-

mental concepts. These exclusions are matters of principle. A third exclusion

is dictated mainly by the ignorance of the author, excused perhaps by the

poverty of the literature, and at any rate violated in several places in the

book: this is extrinsic (combinatorial and differential) geometry or topology.

More than 80% of the material is taken from published papers. The pur-

pose of the notes and bibliography is not to itemize sources but to guide fur-

ther reading, especially in connection with the exercises; so the following

historical sketcn serves also as the principal acknowledgement of sources.

The theory of uniform spaces was created in 1936 by Wei} [W]. All the

basic results, especially the existence of sufficiently many pseudometrics, are

in Weil's monograph. However, Weil's original axiomatization is not at all

convenient, and was soon succeeded by two other versions: the orthodox

(Bourbaki [Bo]) and the heretical (Tukey [T]). The present author is a notor-

ious heretic, and here advances the claim that in this book each system is

used where it is most convenient, with the result that Tukey's system of uni-

form coverings is used nine-tenths of the time.

In the 1940's nothing of interest happened in uniform spaces. But three

interesting things happened. DieudonnS [l] invented paracompactness and

crystallized certain fmportant metric methods in general topology, mainly

the partition of unity. Stone [l] showed that all metrizable spaces are para-

compact, and in doing so, established two important covering theorems

whose effects are still spreading through uniform geometry. Working in

another area, Eilenberg and MacLane defined the notions of category, func-

tor, and naturality, and pointed out that their spirit is the spirit of Klein's

Erlanger Programm and their reach is greater.

The organization of this book is largely assisted by a rudimentary version

of the Klein-Eilenberg-MacLane program (outlined in a foreword to this

book). We are interested in the single category of uniform spaces, two or

v