"I used this book for my first course in topology. It's easy to read and well organized. While it is an introduction, the author covers more than other books in it's price range (Baum or Mendelson, for example). The author starts with some set theory basics and then moves to metric spaces, which he uses as motivation for his definition of a topology (Baum and some others use neighborhood systems instead). He then discusses homeomorphism, seperation axioms (T0 through T4), compactness and other topics. The exercises provide a nice level of challenge; they require thought but aren't impossible. I'd recommend this book as a reference, or to anyone interested in higher mathematics. And since it's a Dover book, the prices is right!"
"I agree with the earlier quite positive reviews: this is a clearly written, concise introduction with plenty of examples. But let's be clear: this book is not really elementary. It's at the level of Munkres Topology (2nd Edition) and requires more mathematical maturity than, say, Mendelson Introduction to Topology: Third Edition, which by the way is an especially reader-friendly first book (but note Mendelson leaves out some key topics, e.g. separation axioms, metrization theorems and function spaces).
Like Mendelson, and unlike Munkres, Gemignani first introduces metric spaces, which are more familiar and intuitive than general topological spaces, because they underlie the usual mathematics in Euclidean n-space we are all familiar with, and then generalizes to general topological spaces. I think this concrete-to-abstract organization particularly lends itself to self-study since people naturally generalize from concrete or restricted cases to general cases. But my very positive evaluation of the book also has a lot to do with the fact that I found Chapter 6 on Convergence superb: it introduces, and provides concrete motivation for, a notion of convergence more general than one based on sequences (suitable for metric spaces) and then develops in some detail, two provably equivalent approaches: the theory of nets and the theory of filters (the latter of which one meets in lattice theory and model theory). (You can also find a discussion of convergence in general topologies in Ch 4. of Willard (1970) General Topology, another earlier book (1970) worth considering.) Those familiar with Isham's Modern Differential Geometry for Physicists (World Scientific Lecture Notes in Physics) might recall that he uses filters in his statement of convergence in general topologies (p. 29 and passim). But Isham's discussion is too abbreviated for one to really understand the material on first exposure: solution - read Chapter 6 Convergence in Gemignani and you're good to go! In contrast, Munkres does not even mention filters, and nets are relegated to Supplemental Exercises (pp. 187-188 in the 2nd edition). Generalized convergence, for some reason beyond me, is simply not treated by Munkres (if you know why or if I missed something, please let me know via a comment to this review.).
A few reviewers have complained about some out-dated notation but personally I hadn't even noticed even though I also have read Mendelson, Munkres and Willard. It is true though that in 1972, when Gemignani was published, terminology in topology had not, as he mentions, been completely standardized, e.g., he does not mention the term "boundary", but he defines "frontier" (p. 55), a term that does not appear in either Mendelson (1975) or Munkres (2nd ed., 1975). Also, there's a very nice section on Identification Spaces (79-85), which would now be called Quotient Spaces (cf. p. 80 and Munkres p. 137). These strike me as minor issues but if your easily frustrated by such differences in terminology or notation, then perhaps a newer book would be a better choice.
In summary: this book deserves careful consideration if you're in the market for a well-written, reasonably complete but concise general topology book with lots of examples that is also very inexpensive. My appreciation for Gemignani has only grown over the years as I have returned to it multiple times for a refresher."
"Gemignani's book explains everything thoroughly and clearly. What's best about it, though, is that the examples are chosen to be do-able and to lock in the material just gone over in the chapter.
Anyone who is a student of higher math and wishes to get a good start on topology will benefit greatly from working through this book, and they will enjoy it too."