Topology Atlas Document # topc-23.htm | Production Editor: Krzysztof Chris Ciesielski

REVIEW OF
Modern analysis and topology
by Norman R. Howes
Univeritext, Springer-Verlag, Berlin and New York, 1995, xxviii+403 pp.,
ISBN 0-387-97986-7

Reviewer

John Mack

Article from volume 2, #2, of TopCom

 

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This book provides a unified development of analysis and topology. These two subjects are studied in the context of uniform spaces. Much of modern analysis is done within the framework of normed linear spaces or of topological groups. Uniform spaces provide the common thread that connects normed spaces and topological groups.

Although the ideas in the introduction and parts of Chapter 1 are elementary and easily within the reach of beginning graduate students, most of this book requires some advanced knowledge. The reader, who does not have the helpful guidance of an instructor, needs to be comfortably at ease in dealing with locally finite covers, with refinements of covers and with similar ideas.

One of the strengths of the book is its description, in Chapter 6, of the very useful and elegant contributions of Professors K. Morita and H. Tamano. This book represents the first time that these results have been assembled together in one place.

Mathematics is blessed with three distinct but equivalent definitions of uniform spaces. They are:

Pseudometrics. This original definition of uniform structures, first published by A. Weil [12] in 1937, requires the existence of a sufficient number of pseudometrics (a requirement that is not always easy to verify). This approach was exploited in Gillman and Jerison [2] and is very useful in functional analysis when the topology on a linear space is given by seminorns. However, this definition is not easily applied to non-abelian topological groups.

Entourages. This definition originated with N. Bourbaki [1] in 1940 and is now the most widely used, by topologists, of the three definitions.

Families of covers. This definition was introduced by J. W. Tukey [11] in 1940 and was used extensively by J. R. Isbell in his book Uniform Spaces [5]. This covering approach is most useful in studying uniform structures that arise in connection with certain topological properties, like paracompactness, that are defined in terms of covers. In this book, Howes uses the covering definition:

A uniform space (X,\mu) consists of a set X and a family \mu of covers of X satisfying the following conditions:
  1. If U and V in \mu then there is a common refinement W in \mu of the covers U and V.
  2. If a cover V of X has a refinement U in \mu, then V in \mu.
  3. Each cover in \mu has a star refinement in \mu.
  4. For every distinct x,y in X, there exists U in \mu so that Star(x,U) and Star(y,U) are disjoint.

For a description of the early development of uniform spaces see [2], page 275 or [8], Notes on pages 186, 207 and 210.

The Introduction lays out the basic facts from general topology while in Chapter 1, the author uses metric spaces to introduce the reader to elementary concepts of uniform spaces such as: uniform continuity, Cauchy sequence, completeness and the completion of a space. The axioms for a uniform space are stated and explicated in Chapter 2.

In Chapter 3, Howes draws the reader's attention to the not so very well known, but very useful, fact that nearly all properties of topological spaces can be characterized via transfinite sequences. Unfortunately, the Ordering Lemma on page 75 is awkwardly stated. Further, the term compatible with is not defined in the book until page 97. A careful reading of the proof of this lemma reveals that the author actually proves the following very interesting fact:

The Ordering Lemma (restated). If a partially ordered set (P,<) is assigned a well ordering <<, then there exists a <-cofinal subset C for which the intersection of CxP with the graph of < is a subset of the graph of the well ordering <<.

When stated in this form, the Ordering Lemma is precisely the tool needed to replaced the convergence of nets with the clustering of transfinite sequences in the characterization of topological properties. Also, the reader may find that a careful look at the (compactified) Tychonoff plank T* as described on page 123 of [2], to be helpful in understanding the Ordering Lemma and in interpreting The Neighborhood Principle on page 76.

The first 4 sections of Chapter 4 describe the fundamental facts about complete uniform spaces; ideas such as: Cauchy net, Cauchy filter, total boundedness, precompact space and the completion of a space. The final section of Chapter 4 fits together naturally with the material in Chapter 5. In this part of the book certain highly technical and very specialized properties of uniform space are taken up. Howes' concept [3] of cofinal completeness and Isbell's [4] supercompleteness are described and contrasted. Also, the technique of lifting a uniformity from a space to its hyperspace, is outlined.

Chapter 6 describes the very interesting work of K. Morita and H. Tamano which uses uniform spaces and product spaces to characterize the following topological properties: realcompact, topologically complete, Lindelof and paracompact. This chapter publicizes these highly useful techniques of Morita and Tamano which are not yet widely know among topologists. Chapter 7 gives an abbreviated description of realcompact spaces. A fuller account may be found in the Gillman and Jerison book [2].

The introduction to measure and integration theory given in Chapter 8 is a prologue to the study of the Haar integral in Chapters 9 and 10. Haar measures are studied in the setting of certain specialized locally compact uniform spaces. In chapters 9 and 10, the work, in this area, of Loomis, Segal, Nacbin, Itzkowitz and others, is described and extended. Howes provides a real service to the mathematics community by putting these ideas in book form. The author exploits the idea of Nachbin [10] of transferring the Haar measure on a locally compact group to a uniform space quotient of that group. The approach used here is to define a special class of locally compact uniform spaces which the author calls isogeneous uniform spaces and then to show that such a uniform space is the uniform quotient of its locally compact group of uniform homeomorphisms. This development paralells that of G. Itzkowitz [7]. (Itzkowitz has, in a private communication to the reviewer, provided a simple, concise {5 line} correction to the error, in his paper [7], which is so prominently mentioned on page 265 of this book.) Also, the claim on page 285: ''Loomis' development contains some errors'' deserves a proper interpretation. Although Loomis' use in [9] of 1950's notation, that is no longer in fashion, makes for difficult reading, the reviewer can find nothing amiss in the Loomis approach.

Chapter 11 deals with function spaces, Lp-spaces and Banach spaces. In Chapter 12, the author introduces the ''uniform derivative'' of a complex measure and then closes the book by proving, under highly restrictive hypotheses, that this derivative is equivalent to the Radon-Nykodym derivative.

The author is to be commended for tracing the history of the important theorems that appear in the book and then giving credit to the original authors of these theorems. However, his efforts in this endeavor sometimes fall short of historically definitive accuracy. A case in point: Theorem 6.16, page 186, is attributed to J. Mack. The reviewer makes no claims of priority with respect to this theorem and is uncertain of its progenitorship. However, its origin almost certainly lies hidden somewhere in the elegant work, during the 1960's, of one or more of the following eminent Japanese mathematicians: K. Morita, T. Ishii, F. Ishikawa or J. Nagata.

In summary, the author does an excellent job of displaying the usefulness of uniform space techniques in the study of both topology and analysis. The highlights of the book are contained in Chapter 3 (transfinite sequences), Chapter 6 (the work of K. Morita and H. Tamano) and the Haar integral in Chapters 9 and 10. Overall, this is an excellent book that deserves a careful reading by the topology community.

References


  1. N. Bourbaki, Topologie générale, Chaps 1-2, Act. Sci. Ind. no. 858, 1940; 3rd ed., Act. Sci . Ind. no. 1142, Paris, 1960. MR 3-55.
  2. L. Gillman and M. Jerison, Rings of continuous functions, D. van Nostrand Company, Inc., Princeton, 1960. MR 22 #6994.
  3. N. R. Howes, On completeness, Pacific J. Math. 38 (1971), 431-440. MR 46 #6303.
  4. J. R. Isbell, Supercomplete spaces, Pacific J. Math. 12 (1962), 287-290. MR 27 #6235.
  5. J. R. Isbell, Uniform Spaces, Math. Surveys, No.12, Amer. Math. Soc., Providence, 1964. MR 30 #561
  6. T. Isiwata, Mappings and spaces, Pacific J. Math. 20 (1967), 455-480. MR 36 #2127.
  7. G. L. Itzkowitz, A characterization of a class of uniform spaces that admit an invariant integral, Pacific J. Math. 41 (1972), 123-141. MR 47 #7001.
  8. J. L. Kelley, General Topology, D. van Nostrand Company, Inc., Princeton, 1955. MR 16-1136.
  9. L. H. Loomis, Haar measure in uniform structures, Duke Math J. 16 (1949), 193-208. MR 10-600.
  10. L. Nachbin, The Haar Integral, D. van Nostrand Company, Inc., Princeton, 1965. MR 31 #271.
  11. J. W. Tukey, Convergence and uniformity in topology, Princeton University Press, Princeton, NJ, 1940. MR 2-67.
  12. A. Weil, Sur les espaces à structure uniforme et sur la topologie générale, Act. Sci. Ind. no. 551, Paris, 1937.

This document was last modified on April 29, 1997.