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Introductory Topology: Exercises and Solutions

Mohammed Hichem Mortad
World Scientific
Publication Date: 
Number of Pages: 
Problem Book
[Reviewed by
Mark Hunacek
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This is a problem book in (very) basic point set topology. I don’t see this as a text from which a student can actually learn the material, but, subject to some limitations (discussed below), this book may serve as a supplemental text.

The book is divided into two parts, “Exercises” and “Solutions”. Each chapter in the first part (except for chapter 1) is divided into five sections, following the same format. The first section (“What You Need to Know”) summarizes the basic definitions and theorems (statements only, no proofs). The next section is entitled “True or False: Questions”, and consists mostly of true-false questions, although occasionally a non-declarative question that cannot be so answered slips in. (Example: “How do we know that a given metric space is complete?”). The third section (“Exercises with Solutions”) lists a number of exercises, followed by more (generally shorter) exercises in section 4 (“Tests”). A final section (“More Exercises”) provides additional problems, solutions to which are not available in the text. Part II of the text is devoted to solutions to the exercises appearing in sections 2–4 of the chapters in Part I. (The questions in the “Tests” section of each chapter are not given the same kind of detailed solutions as are the ones in the “Exercises with Solutions” section.)

It bears emphasizing that this is not a “Moore method” or “inquiry based learning” text. In those kinds of texts, the student is carefully led through a series of problems to discover the basic underlying results of the theory for himself or herself. Here, the basic theorems of the subject are simply given to the student, who may take them as assumed; the assumption is that the students will have seen proofs of these in another text or in a lecture, and the author explicitly notes in the preface that this section cannot replace a more detailed course. The student is then given a bunch of exercises to solve, much like the exercises at the end of a section in a traditional text. So, this book is, I think, best viewed as a supplement to another text, rather than as a text itself.

I referred, in the opening paragraph of this review, to some limitations of this text. The first is its fairly narrow scope of coverage. The book begins, after an introductory chapter on sets and functions, with a chapter on metric spaces (chapter 2), followed by one on topological spaces (chapter 3). These are followed by chapters on continuity and convergence, compactness and connectedness (chapters 4–6, respectively), after which the book concludes with chapters on complete metric spaces (chapter 7) and function spaces (chapter 8). In each of these chapters, the discussion does not get terribly deep, and largely tests understanding of the basic definitions and concepts.

As a result, there are a number of topics that might be covered in an undergraduate course that are simply absent here. Some texts, for example, at least mention the fundamental group; Mendelson’s Introduction to Topology, for example, devotes two sections of the chapter on connectedness to the fundamental group and the concept of a simply-connected space. This book doesn’t mention the concept at all. This is reasonably understandable, since algebraic topology is still not a common part of an undergraduate course, but other omissions are more puzzling. The concept of a quotient space, for example, is also completely omitted, and consequently we don’t see any of the interesting examples that accompany that idea, such as Möbius strips and Klein bottles. I also don’t recall seeing any reference to normal or completely regular spaces, and the idea of first and second countability shows up only briefly. The only products that are considered are finite.

A comparison of the topics in this book with the topics covered in Conway’s A Course in Point Set Topology (which is already a fairly stripped-down text) shows that about three quarters of the material in the latter book is represented here. I’m not quite sure why the author didn’t take the opportunity to explore some more topics; the book as written is fairly short (less than 250 pages) and could easily have accommodated another 75 or 100 pages. Readers who view these omissions as significant can peruse Elementary Topology: Problem Textbook by Viro et al., which is substantially longer (400 pages) and which covers a considerable number of topics not found here.

Another limitation concerns the exercises themselves, most of which are quite routine. A substantial majority of them involve working with specific examples, and ask the reader, for example, to show that some collection of subsets is or is not a topology, that a given function is a metric, to determine whether some subset of a topological space is open or closed, to identify the closure or boundary of a subset, and so on. Some exercises do call for proofs, but these are generally very simple (e.g., prove that a finite topological space is compact). Occasionally a substantial problem (e.g., asking the reader to show that the topologist’s sine curve is connected but not path-connected) shows up, but when this happens, it is often in the “More Exercises” section of the chapter, so no solutions are forthcoming.

The exercises that appear here with solutions are very much like the ones that would be expected to appear at the beginning of an end-of-section exercise set in a textbook. These problems may serve well as a routine drill for students seeking to cement their understanding of the basics, but it does seem like a missed opportunity to omit some more difficult problems, especially since there are some fairly famous ones, such as the Kuratowski closure-complement problem: starting with an arbitrary subset A of a topological space X and successively taking complements and/or closures in any order, show that at most 14 different sets can be obtained. This old chestnut appears as an exercise in some undergraduate texts, such as Munkres’ Topology (which also asks for an example of a set where the number 14 is obtained); it would be nice to have a convenient reference to a solution.

On the plus side, there are some attractive features of the book. The solutions to the section 3 “Exercises with Solutions” questions are fairly detailed, and thus, notwithstanding some occasional idiosyncratic English (in one question, for example, the author, wanting the student to identify the derived set A' of a set A, asks “What is A' worth?”) a student reader should have little trouble getting though this book without much or any assistance. I also thought the idea of numerous true-false questions was a particularly good one; I don’t see questions like this very often in undergraduate topology textbooks, and my guess is that they could prove valuable in firming up a student’s understanding of the material and ability to construct counter-examples. (One question that I thought was especially valuable was whether the statement “completeness is a topological property” is true or false; decades after the fact, I still recall being embarrassed by my naïve assumption, as an undergraduate, that the answer was true.)

So, on balance, this book’s objective of being a resource for introductory students learning topology for the first time has, I think, been met, though I do wish there was a bit more coverage and some more difficult solved exercises. In addition, although the book was written for a student audience, it seems clear that it should also prove quite valuable to instructors of an introductory topology course, as a readily available source of supplementary homework questions or exam questions.

Mark Hunacek ( teaches mathematics at Iowa State University. 

  • Exercises and Solutions:
    • General Notions: Sets, Functions et al.
    • Metric Spaces
    • Topological Spaces
    • Continuity and Convergence
    • Compact Spaces
    • Connected Spaces
    • Complete Metric Spaces
    • Function Spaces