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