Elementary Topology: Second Edition (Dover Books on Mathematics)

It's is a good book on the elementary book of topology. The author does a good job explaining the materials well. Only wish he put more examples in each section. If you don't know topology this is a good start. Even thou it is an older textbook the material is relevant today.

The best introductory book on topology ever!

definition, proof, definition, proof, definition, proof,.....

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.

This book is only helpful for those who already have a solid foundation in Topology. It is a good quick-reference guide but the explanations are too short to provide any self-teaching capability. This book has no solutions to problems and most of them are ambiguous proofs with limited examples to help. Graphs and pictures are limited. I would expect books for math theory to have plenty of visualizations instead of expecting students to understand the overlapping and increasing jargon on every page.

the book follows the straightforward style of proposition-theorem-proof-corollary and full of examples and helpful diagrams. but the notation is not fully compliant with modern topology.

É um excelente livro de topologia. Trás os assuntos básicos da topologia de uma maneira simples, porém trata-os com o rigor necessário. É um bom livro para se aprender e reforçar o aprendizado da topologia geral e aprender os rudimentos da topologia algébrica.

Un testo meraviglioso, formale, completo, veramente un gioiellino da studiare dalla prima all'ultima pagina!

I was interested in developing Topology without Metric Spaces. This book does it in a clever way using Metric Spaces just to clarify, by examples, the general Topology ideas.Great book. I will use it to formalizing Topology from ZFC.