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Convergence Foundations of Topology

Szymon Dolecki and Frédéric Mynard
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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To quote Robert Frost, this text takes a road less travelled to get to at least one of its destinations, viz. the fundamentals of topology that everyone should know. Well, perhaps this evaluation needs to be qualified: the authors claim in the introduction to their book that they offer this text as a first salvo in the cause of teaching analysis — that’s their bigger picture. So it can be argued that the road they take is a bit too parochial. On the other hand, the emphasis they place on convergence as such is easily defended on the count that this analytic notion is certainly ubiquitous in mathematics, consonant with the fact that analysis is literally everywhere. But then again, so are open sets and closed sets, so I guess the jury remains out on this count. Still, the authors bolster their case by pitching the book to “senior undergraduates getting familiar with topology for the first time, graduate students seeking a deeper understanding of the foundations of analysis, as well as experienced mathematicians.” The authors then go on to recommend that in teaching from the present book the instructor pick and choose as a function of the needs of his audience , the three classes just mentioned being suggestive.

Explicitly, Dolecki and Mynard note that “[t]he book also serves as a useful reference … [even] for topologists, as it gathers for the first time in a textbook format a wealth of basic results and examples on convergence spaces.” So we have just encountered an interesting fauna along this less trodden path: what is a convergence space? By definition (cf. pp 55–56 of the book), it is given by a set \(X\) and a “convergence \(\xi\), which is characterized in terms of filters on \(X\), i.e. we have ab initio a notion of a filter converging to a point of the space, and, yes, we are dealing with the corresponding notion from the theory of ordered sets. To wit, a filter on \(X\) (cf. pp. 22, 29) is a family of subsets of \(X\) containing the empty set, closed under supersets, and closed under intersections; and then the notion of the limit of a filter with respect to a relation \(\xi\), from filters to \(X\), is presented in a rather austere manner: it can’t really be otherwise in the world of filters and ultrafilters.

It is perhaps fair to say that this increase in austerity due to such an abundance of set theory of a rather underappreciated type is balanced by the depth of understanding one acquires by going this route: the student (and reader) certainly comes away from all this with a much expanded view of what convergence is, aside from, or even before, the explication of the notion in such a prosaic context as the analysis situs of the real line. There are at this stage no epsilons and \(n\)s or deltas, but there is a set-theoretic sense of passing through a filter to get to a limit.

For people like me, whose understanding of convergence is steeped in the classical approaches going back to Cauchy and Weierstrass, the present perspective is indeed surprising and fascinating. The authors manifestly achieve their goal of expanding our horizons so as to include the according revolutionary work done by Claude Choquet — cf. the Historical Remarks on pp. viii–ix: I was surprised to learn that the standard notions of open sets and closed sets, of interiors and closures of sets, go back to Cantor and Peano, for example; beyond this Dolecki and Mynard mention Frechet, Hausdorff, Vietoris, Kuratowski, Čech, and Sierpinski as the major figures in developing the theory of convergence in metric (or metrizable) spaces. The “conceptual turnover” came with Choquet, dating to 1947–1948, but the notion of filter goes back to Henri Cartan, ca. 1937, with foreshadowing by none other than Leopold Vietoris. So we do see the natural connection with topology: some of the founders, themselves, were in the game.

Back to the book under review. Where do we encounter the landmarks of basic topology, travelling the land by this other road? Well, it isn’t until Chapter IX that we encounter compactness, and then in the following form (cf. p. 231): “A convergent space is compact if every filter on its underlying set is adherent, equivalently if each ultrafilter on its underlying set is convergent.” We are in fact in familiar territory, but we’re approaching it from another direction: on p. 234 we get Bolzano-Weierstrass in the following form: “A subset of \(\mathbb{R}\) is compact … if and only if it is closed and bounded; interestingly (and reassuringly) the appended proof is wrought with familiar features. And on it goes. On p. 236 we get that the continuous image of a compact set is compact, for example. It is in this chapter, moreover, that we witness one of the primary virtues of this book, namely, that having built such a ramified set-theoretic foundation, many more themes are available than one generally encounters in topology at an early stage. Indeed, this chapter is filled with deep and important themes, such as the Stone topology and a very detailed comparison of countable compactness, sequential compactness, and compactness proper. Finally, on p. 275 we read: “Compactness can also be expressed in terms of covers,” and in short order we learn that for a topological space to be compact it is indeed necessary and sufficient that every open cover have a finite subcover, and we are home again.

The next chapter deals with completeness in metric spaces and starts off with a discussion of the notion of a Cauchy or fundamental filter, bringing to mind the more familiar “competing” notion of a Cauchy or fundamental sequence, of course, so it is possible to say that at this stage the authors’ approach is beginning to converge, morally, to the standard one (if I may be excused an egregious pun). And things continue in this way: the next chapters deal with connectedness and compactification.

On the other hand, the book’s last handful of chapters are again somewhat non-standard, at least as far as opening courses in topology go, dealing as they do with, e.g., classification questions and notions of duality that are pretty esoteric, at least from a mainstream topology/analysis perspective.

The very first sentence of the authors’ Preface is this: “The aim of this book is twofold: an elementary original introduction to topology and an advanced reference on convergence theory.” It is true, indeed, that the book is original, and it looks to be a fine reference on convergence theory at a very high level. But the use of the word “elementary” needs qualification: yes, the material is built up from its foundations in set theory, but it requires considerable mathematical sophistication from the audience. It is clear that the book, with its high level of scholarship, is a labor of love on the part of its authors, and it will no doubt make quite a dent in the right circles. But the novice had better be able to pick and choose and navigate carefully along these roads. The mature reader, disposed to this way of redoing topology, will have a great time.

Michael Berg is Professor of Mathematics at Loyola Marymount University  in Los Angeles, CA.

  • Introduction
  • Families of Sets
  • Convergences
  • Continuity
  • Pretopologies
  • Diagonality and Regularity
  • Types of Separation
  • Pseudotopologies
  • Compactness
  • Completeness in Metric Spaces
  • Completeness
  • Connectedness
  • Compactifications
  • Classification of Spaces
  • Classification of Maps
  • Spaces of Maps
  • Duality
  • Functional Partitions and Metrization
  • Set Theory