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Metric Spaces

Mícheál Ó. Searcóid
Springer Verlag
Publication Date: 
Number of Pages: 
Springer Undergraduate Mathematics Series
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Marion Cohen
, on

On page 245 of this book the author says, “The theorem-proof approach to the writing of mathematics is followed throughout…” This to me is good news. I recall graduate school topology class; the professor stood at the board and, with what to me seemed a deadpan face under a subtle smirk, simply wrote down, in minimal notation, what he had to tell us. He wrote T. for theorem, P. for proof, and, at the end of every proof, a huge alpha instead of Q.E.D. He offered no motivation, no explanation other than the very clear definitions, theorems, and proofs — nothing, that is, except the math itself. And I was enthralled!

This book does, however, contain plenty of both motivation and explanation — right along with “theorem-proof”. So in this book we have the best of both worlds. There are also many interesting examples, notes, and questions. (So It’s not exactly “theorem-proof”, but theorem-proof-examples-notes-questions. But yes, each paragraph is titled “theorem” or “proof” or “example”, etc, and is also numbered.)

Motivation, explanation, and the author’s own personaility survive throughout. For example, on page 181, “There is no prize for guessing what the unique fixed point of this contraction is.” And page 201, about space-filling curves, “Therefore S, despite all appearances, is the image of a path — it is an example of what we are hesitating to call a curve.” Also engaging are his several Venn diagrams shaped like hearts and his very-own cartoon drawing, on p. 249, of the domino effect in the case where the dominos are set up to form a closed curve beginning and ending with the character who is proudly about to push the first domino. (What or who will the last domino push?)

This book is truly about metric spaces. Topological spaces are thoroughly mentioned, as are theorems which apply only to metric spaces but not to topological spaces in general. But metric spaces are the emphasis; we feel the distances, and we visualize the resulting concepts — distance between sets (as well as between points, and from points to sets), boundaries, open, closed, and dense sets, balls, convergence, continuity, uniform and Lipshitz continuity, completeness and compactness, connectedness, and when and how two metrics might be comparable (in various ways) to each other. I think that students will appreciate, as I did, the opporunity to fully explore the notion of “generalized distance”, with lots of examples both in agreement with and in slight disagreement with intuition.

The book is packed full of material which does not often appear in comparable books. Pointlike functions, the Hausdoff metric, the Axiom of Dependent Choice (in the appendix), certain theorems about chained collections of sets (also in the appendix, but used in the text itself), and something called “conserving metrics”, a term which he coined to ease our way through some of the theorems and proofs and to, in his words (p. 238) “smooth the reader’s path to understanding”. In my opinion, this last is commendable. I’m always impressed and delighted when a textbook writer does non-expository writing as well.

Other commendable characteristics about the book include the numerous passages (e.g., the top of p. 234) where, even though he’s at a more advanced part of the book, he refers back to very elementary (though not trivial) material, as well as passages where he proves more than he needs to, just for the heck of it. His use of questions to increase understanding and to move on to the next topic are also to be appreciated. Also, he keeps abreast of pitfalls that could confuse beginning students. (E.g., p. 103: “The terms unbounded and infinite tend to be used in common parlance as if they meant the same thing. Mathematically, they are quite different.”) Finally, on p. 127 appears an example of his welcome tendency to introduce, early on, concepts which will later connect with more advanced concepts; he asks, in effect, what can be said about the ratio between (the “largest”) delta and each epsilon? This is interesting in itself, and also gets its day in court when he introduces Lipshitz functions, and investigates their derivatives (p. 157). And it’s cool when he asks, in his word, “strange” questions like (p. 133) “Are metrics continuous?”

Now for some nitpicks — things that I would do differently (not necessarily better…) if it were my book: I would, when describing small open balls, use epsilon rather than r (though perhaps he wanted to save epsilon for continuity). And I would say, when appropriate, “for all epsilon, no matter how small “. Sometimes he uses the notation x --> d(z,x) to indicate a function, when it seems clearer to simply say δz(x) = d(z,x).Also, it might be good to give, especially towards the end of the book (e.g., Section 12.9 on paths of minimum length), more descriptions of the essence of some of the theorems and proofs, in particular those which are more involved (and not just what I call “definition and theorem hopping”). (For the last statement in Theorem 12.9.7 I would say “the path g moves along uniformly — always at the same “speed”.). Finally, I disagree with his accessment of “iff” (to mean “if and only if”) on page 246 in Appendix A. (That is, I don’t believe that “iff” is a “horrible” abbreviation.) .

A couple of times, if I’m correct, he gets carried away, meaning that he proves what he needs to prove but then keeps going (seeming to forget what it was that he was aiming to prove in the first place — easy to do!). E.g., On p. 135 he tells us that, in a metric space, any two non-empty disjoint closed subsets can be separated by a continuous function, and that he will show us how. This he aptly does, but then he goes on to show something that we already knew (that they can be separated by disjoint open sets). I also would treat the completion of metric spaces slightly differently (p. 187). I would define “virtual points” to include the (actual) points of X and then define the completion to be the set of virtual points. (Perhaps I’d also change the term “virtual points” to “generalized points”.) And on p. 223, the sequence of sentences and equations in the proof of the theorem which leads to our being able to call lp spaces normed spaces should, I think, be re-examined; a slight permutation would improve the clarity.

There seem to be a small number of typos — e.g., on p. 139, line 1, I’m sure the author meant “subsets of X”. On p. 212 “finite” is left out, and on p. 209, “uniformly” is left out. And on p. 56, Question 4.1.12, lines 7–8, I think he must mean “is nonetheless closed”, rather than what is there.

Finally, if I may, I have a small suggestion, a mneumonic device which helped me, at least, with the meaning of his coined term “conserving metric”: anything between suP and suM.

All-told this is a great book and suitable, as it claims, for third-and fourth-year under- graduates and beginning graduate students. In the language of metric spaces, topology, and humanity, it is open (to further exploration), closed (meaning self- contained), and dense.

Marion D. Cohen has a poetry book in press, forthcoming from Plain View Press ( ), about the experience of mathematics. The title of the book is “Crossing the Equal Sign”.  She would love to receive emails at: [email protected]

 To the Reader.- Cumulative Reference Chart.- Metrics.- Distance.- Boundary.- Open, Closed and Dense Sets.- Balls.- Convergence.- Bounds.- Continuity.- Uniform Continuity.- Completeness.- Connectedness.- Compactness.- Equivalence.- Appendices: Language and Logic.- Sets.- Solutions.- List of Symbols.- Index.