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Introduction to Topology
Highly regarded for its exceptional clarity, imaginative and instructive exercises, and fine writing style, this concise book offers an ideal introduction to the fundamentals of topology. It provides a simple, thorough survey of elementary topics, starting with set theory and advancing to metric and topological spaces, connectedness, and compactness. 1975 edition.
224 pages, Paperback
First published January 1, 1975
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Displaying 1 - 22 of 22 reviews
December 14, 2013
I'm not yet enough of an expert on topology to give this a truly informed review; I can compare this to only to one other topology text (but plenty of math books).
The book starts out with a very compact review of set theory (no pun intended). Then comes a warmup chapter on metric spaces, which seems a bit cumbersome in places, but is turns out to be carefully designed to use exactly the same structure as will be used when introducing similar topics for topological spaces. The tour through metric spaces is quite helpful, introducing abstract definitions of topics such as continuity, neighborhoods and limit points that are crucial in topological spaces while the reader still has the crutch of distances to fall back on to make sense of the concepts. The topology content covered is entirely point-set topology, covering the basics of openness, compactness, and connectedness, without moving any deeper.
The book is very self-contained, and should be accessible to any undergrad comfortable with proofs and preferably at least some exposure to real analysis. That's not to say reading it is simple, however; expect to put in some work following the proofs and solving problems if you want to get something out of it.
The style of the book is quite elegant and spare. It doesn't elaborate once an argument or proof is complete, but is quite readable despite its terseness. This style will be too brief for students who want more redundancy, hand-holding, and worked exercises. But it's quite enjoyable for those who are willing and able to supply the mental effort and let the author guide your efforts. The book includes plenty of end-of-section examples, of a range of difficulties. Working a sampling of these after each section is plenty to ensure that you have understood the material. One thing I would have appreciated, though, is a few more examples of non-Euclidean topological spaces that I could have used as exercises to poke at the definitions and proofs. My intuition is still embedded way too deeply in Euclidean space, and I have to stress-test it to know when I'm relying on it too much.
Some of the terminology used is slightly non-standard, or out-of-date. The theorems never change, but the wording does, slowly, and this is visible when Mendelson uses terms like identification topology rather than quotient space. This makes it slightly more inconvenient to match topics when supplementing this text with other sources or vice versa, but I suppose this is inevitable in a book that was first published more than 50 years ago. Texts in fields other than math tend to become dated much faster.
The best part about the book is the price. Dover editions are nearly always a bargain, and this one is no exception. It almost seems like it should be illegal to get this much good stuff for $10.95.
The book starts out with a very compact review of set theory (no pun intended). Then comes a warmup chapter on metric spaces, which seems a bit cumbersome in places, but is turns out to be carefully designed to use exactly the same structure as will be used when introducing similar topics for topological spaces. The tour through metric spaces is quite helpful, introducing abstract definitions of topics such as continuity, neighborhoods and limit points that are crucial in topological spaces while the reader still has the crutch of distances to fall back on to make sense of the concepts. The topology content covered is entirely point-set topology, covering the basics of openness, compactness, and connectedness, without moving any deeper.
The book is very self-contained, and should be accessible to any undergrad comfortable with proofs and preferably at least some exposure to real analysis. That's not to say reading it is simple, however; expect to put in some work following the proofs and solving problems if you want to get something out of it.
The style of the book is quite elegant and spare. It doesn't elaborate once an argument or proof is complete, but is quite readable despite its terseness. This style will be too brief for students who want more redundancy, hand-holding, and worked exercises. But it's quite enjoyable for those who are willing and able to supply the mental effort and let the author guide your efforts. The book includes plenty of end-of-section examples, of a range of difficulties. Working a sampling of these after each section is plenty to ensure that you have understood the material. One thing I would have appreciated, though, is a few more examples of non-Euclidean topological spaces that I could have used as exercises to poke at the definitions and proofs. My intuition is still embedded way too deeply in Euclidean space, and I have to stress-test it to know when I'm relying on it too much.
Some of the terminology used is slightly non-standard, or out-of-date. The theorems never change, but the wording does, slowly, and this is visible when Mendelson uses terms like identification topology rather than quotient space. This makes it slightly more inconvenient to match topics when supplementing this text with other sources or vice versa, but I suppose this is inevitable in a book that was first published more than 50 years ago. Texts in fields other than math tend to become dated much faster.
The best part about the book is the price. Dover editions are nearly always a bargain, and this one is no exception. It almost seems like it should be illegal to get this much good stuff for $10.95.
July 2, 2012
You caught me--technically, I haven't finished this book yet. I'll admit that I'm not quite ready to tackle topology itself. However, the introductory overview of set theory and set theoretical notation is invaluable if you're interested at all in foundational mathematics or philosophy of mathematics (be it Russell or Badiou). I had worn myself out on other books dedicated to the subject, and this book provides a clearer, more to-the-point account of basic set theory than any I've found (in a fifty page introduction, no less!).
In this regard, it's worth the read. I'll get back to you about the rest when I'm more prepared to treat topology in earnest.
In this regard, it's worth the read. I'll get back to you about the rest when I'm more prepared to treat topology in earnest.
April 20, 2015
I have the patience of a hummingbird. Therefore I`m well-equipped to appreciate any popularly pitched math textbook. An Introduction to Topology is a consistent, accessible, and charming compendium of a few higher branches of symbolic maneuvering. I dare you to read it.
June 17, 2018
I was expecting something more readable. This book seems very difficult for an introduction. Therefore, I am going to buy another books about this subject and then compare them with this one.
September 6, 2009
I actually have the Second Edition,( must speak with a librarian or become one to add this and several other 'older' books )of this standard. Not unlike several other 'Intro' books, there is a rather large amount of pre-req knowledge required. If you are firmly rooted in your understanding of Set Theory and have had rigorous levels of experience with that nomenclature, only the first couple of readings will be a muddle.
One of the best formal introductions to basic Topology. If you are studying any of the modern flashback theories of the universe, the relationship between Hilbert Space and Euclidean inclusion along convergences is a must!
A little dense initially on the important topic of convergence, but you will get there more quickly than with most other elaborations that I know of up to books even as recent as 4-5 years ago(circa 2009). This is basic stuff folks! Get it right the first time and your wonderment over dimensionality will become a diminishing metric subspace (oops, another obscure application of Topology and not intro level).
Recommended.
One of the best formal introductions to basic Topology. If you are studying any of the modern flashback theories of the universe, the relationship between Hilbert Space and Euclidean inclusion along convergences is a must!
A little dense initially on the important topic of convergence, but you will get there more quickly than with most other elaborations that I know of up to books even as recent as 4-5 years ago(circa 2009). This is basic stuff folks! Get it right the first time and your wonderment over dimensionality will become a diminishing metric subspace (oops, another obscure application of Topology and not intro level).
Recommended.
September 25, 2018
Good lord I hate topology. Good book on it, though.
August 14, 2023
This is a really strange textbook!
So the content (by UK-equivalent standards) is mostly GCSE/A-Level Mathematics content as it places a massive emphasis on set theory, and digs very deep into topology through set theory at the expense of everything else - including novel/interesting teaching - making this textbook a set-theoretic topology textbook and not a proper introduction to topology.
Mendelson simply throws you under the bus of intensive set theory, and offers nothing else for understanding topology. So if you're, like me, a topologist who prefers visual intuition; examples/counterexamples; practical problems; and a varied series of methods, then this book is not for you.
Even with a strong understanding of set theory, I found this textbook to be insanely dreary and learnt almost nothing new. I will however, give it a second chance, and return to it after starting a lecture series on topology to see if my attitude towards the textbook changes.
However, if you enjoy set-theoretic topology - and are looking for a textbook with hundreds of exercises (which don't really test anything other than your understanding of sets) - then this is the book for you.
People looking for looking at cool non-Euclidean manifolds and working with them mathematically - look elsewhere!
So the content (by UK-equivalent standards) is mostly GCSE/A-Level Mathematics content as it places a massive emphasis on set theory, and digs very deep into topology through set theory at the expense of everything else - including novel/interesting teaching - making this textbook a set-theoretic topology textbook and not a proper introduction to topology.
Mendelson simply throws you under the bus of intensive set theory, and offers nothing else for understanding topology. So if you're, like me, a topologist who prefers visual intuition; examples/counterexamples; practical problems; and a varied series of methods, then this book is not for you.
Even with a strong understanding of set theory, I found this textbook to be insanely dreary and learnt almost nothing new. I will however, give it a second chance, and return to it after starting a lecture series on topology to see if my attitude towards the textbook changes.
However, if you enjoy set-theoretic topology - and are looking for a textbook with hundreds of exercises (which don't really test anything other than your understanding of sets) - then this is the book for you.
People looking for looking at cool non-Euclidean manifolds and working with them mathematically - look elsewhere!
August 14, 2018
This is my first book on Topology, and it's quite good. First few chapters are basic and about sets, functions. The nest thing I liked about this book was that it connected the topics connectedness to intermediate value theorem and as such. It was sort of helpful in getting the big picture of Topology and its relationship to calculus and other branch of Mathematics. Great Book to start studying Topology
June 28, 2018
It has taken me roughly 20 hours to read the first 40 pages. That was 4 years ago. Last few weeks I have worked through the rest in maybe 10 hours.
What made this book stand out for me back then eas, that it was very indroductory and didn't assume much. The exercises have clarified the material and have been fun to work through.
What made this book stand out for me back then eas, that it was very indroductory and didn't assume much. The exercises have clarified the material and have been fun to work through.
May 8, 2022
Did a decent job introducing the initial concepts, but never gave me any motivation nor payoff and I found it difficult to care about just how open or closed various spaces are or are not on their own merits.
September 19, 2017
Good book but very dense for a non-mathematician like myself. Doesn't really help one develop an intuition of the subject.
September 9, 2021
Primarily algebraic in orientation, it concludes with the fundamental group of topology. Pretty good overall introduction to topology, short and concise.
November 27, 2021
If you want a really well written introduction to topology with pretty interesting problems to practice, start from this one before going to Munkres.
January 19, 2023
Starts with set theory, moves into metric spaces and builds from there. Probably as good an introduction to the topic as you're going to get.
May 24, 2023
Very good
April 23, 2025
One I'll be revisiting
January 2, 2016
Overall, great introductory book to topology. The pedagogy was excellent and the development of topics made sense in going from metric spaces (a notion that is generally more intuitive) to abstract topological spaces.
In particular, it was great for self-study as Mendelson doesn't shy away from fully fleshing-out proofs and repeating relatively similar cases with some additional notes (e.g. when going from metric to topological spaces and proving several ideas there). The book itself can certainly be read by anyone with a set theory background and some intuitive notion of limits/sequences (i.e. a class in pre-calculus), but that doesn't mean it's easy, by any means. I struggled quite a bit with the intuition behind some of the proofs, and have, more than once, rolled around on my bed trying to recall (or prove again) some particular statement that I found quite useful. Sadly, the book doesn't have a section on homotopy equivalence and some other useful notions, but do recall it is an introduction in exactly 200 pages of short text.
This book took me at least 20-30 hours to get through, skipping only the very latter section of the compactess chapter and doing at least two of the harder problems in each section; but I have very little experience with analysis, something I'm sure would have helped complete this and gain the corresponding intuition much more quickly.
Again, great book which I would highly recommend for self-study.
In particular, it was great for self-study as Mendelson doesn't shy away from fully fleshing-out proofs and repeating relatively similar cases with some additional notes (e.g. when going from metric to topological spaces and proving several ideas there). The book itself can certainly be read by anyone with a set theory background and some intuitive notion of limits/sequences (i.e. a class in pre-calculus), but that doesn't mean it's easy, by any means. I struggled quite a bit with the intuition behind some of the proofs, and have, more than once, rolled around on my bed trying to recall (or prove again) some particular statement that I found quite useful. Sadly, the book doesn't have a section on homotopy equivalence and some other useful notions, but do recall it is an introduction in exactly 200 pages of short text.
This book took me at least 20-30 hours to get through, skipping only the very latter section of the compactess chapter and doing at least two of the harder problems in each section; but I have very little experience with analysis, something I'm sure would have helped complete this and gain the corresponding intuition much more quickly.
Again, great book which I would highly recommend for self-study.
October 25, 2009
ugh. this be difficult.
November 28, 2012
Excellent resource for learning about topology, although I wasn't really prepared for several proofs and exercises so it was hard to wrap my mind around some concepts. Overall a good book.
March 18, 2013
Good, but very quick and straight to the point. You will find maybe a paragraph or two of intuition. This book helped me immensely in analysis.
September 21, 2009
Not well written in my view, but serves an an intro for use with diff. equations.
Displaying 1 - 22 of 22 reviews