"Schaum's Outline of General Topology" by S. Lipschutz. Is it good choice for self study General Topology? I am looking for a book to self study general topology. I found "Schaum's Outline of General Topology"  by S. Lipschutz. Is it good choice? Many people recommend other books (for example Munkres "Topology") but "Schaum's" Topology is very interesting. There are many examples with solutions. I like this book but I have doubts because not many people recommend it, and for example is written in 1965 (maybe not cover all material, or don't have good modern mathematical style). Is it good choice for self study?
Thanks, Zbynek
PS Sorry for language mistakes.
 A: I have used both Schaum's General Topology and Munkres, and think that Schaum's is a great choice!  Both have their advantages, so let me elaborate a little below:


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*Exercises: Both Schaum's and Munkres have plenty of exercises, but the solutions in Schaum's are very helpful.  Further, I would say that for self-study Schaum's has the edge, since it has more exercises that fall in the "easy" range.  These are the kind of exercises that help you really get comfortable with new mathematical objects and ideas.  Munkres has some of these problems too, but I would say that the distribution is more weighted towards problems that develop new, and sometimes challenging material.

*Material: Both Schaum's and Munkres cover roughly the same material, with the exception of Algebraic Topology.  Munkres is divided into two sections: the first is general topology, and the second is Algebraic Topology.  Schaum's covers roughly the same material as in the first section of Munkres but doesn't have the second section at all.  If you're interested in going beyond what most people would consider a standard introduction to general topology, then go with Munkres.  That being said, I think it might be best to go with Schaum's and really learn the basics well, then choose a book that's specifically about Algebraic Topology.
Hope this helps, and you really can't go wrong with either - they're both great books for someone who's new to the subject.
A: Althought I am not a fan of Schaums Outline, it might have some advantages (step-by-step examples) and so on. But let me give some alternatives.
Let me start with some full PDFs that are publicly available (where you might judge directly whether it is suitable for you or not):

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*A Textbook of Topology (Elsevier)

*Topology and Geometry (Springer)

*Schaum's Outline of Theory and Problems of General Topology (it is the subject of the question)

*Topology without Tears (mentioned in the comments)

*Introduction to Topology (Lecture Notes, Colorado State University)

I can recommend these books in good conscience if you want to get started in topology:

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*From Geometry to Topology (H. Graham Flegg) provides a very gentle introduction for self-learners. It starts with congruence classes and explains everything geometrically and later it introduces key terms such as "connectivity" and also here provides geometric meanings. You may miss the algebraic part and find it a bit too simple. But I prefer to get abstract things envisioned.

*Geometry and Topology (Miles Reid, Balázs Szendrői) is a Cambridge book and goes further - but in a similar gentle way. It starts with Euclidean geometry and introduces metrics, spaces and motions (chapter 1). Then it introduces Composing maps (chapter 2), Spherical and hyperbolic non-Euclidean geometry (chapter 3), Affine geometry (chapter 4), Projective geometry (chapter 5), Geometry and group theory (chapter 6), Topology (chapter 7), Quaternions, rotations and the geometry of transformation groups (chapter 8) and concluding remarks (chapter 9). The chapter "Topology" itself introduces the definition of a topological space and motivates it from metric spaces. Moreover terms such as continuous map, homeomorphism etc. and topological properties are explained. There are many illustrations/figures and each chapter ends with exercises. It is a great book for self-learners. The book does not provide solutions for the exercises (but you will become resourceful here in the community).

Here is an example (cut out) of the exercises:


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*An Illustrated Introduction to Topology and Homotopy (Sasho Kalajdzievski) is another good book for self-learners. It is a CRC (Taylor & Francis) book and I find most of CRC books for self-learners very nice too. It has two parts: Topology (Part 1) and Homotopy (Part 2). The first part dealth with Sets, Numbers, and Cardinals (chapter 1), continues with Metric Spaces, Topological Spaces (chapter 2 and 3), Subspaces, Quotient Spaces, Manifolds, and CW-Complexes (chapter 4), Products of Spaces and Connected Spaces and Path Connected Spaces (chapter 5 and 6), Compactness and Related Matters (chapter 7), Separation Properties and Urysohn, Tietze, and Stone–Čech (chapter 8 and 9). This book goes deeper than the two previous ones. It provides a lot of proofs, examples and also exercises. What I like really much is that this book references the excercises in examples, proofs and texts consistently. Solutions are not given, however in the exercises often the related sections and examples are linked. Last year the Solutions Manual for Part 1 Topology has been published by Sasho Kalajdzievski, Derek Krepski and Damjan Kalajdzievski. This solution manual utilizes very helpful illustrations as well that simplify the comprehension of exercise solutions. Here is one example:



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*Algebraic topology (Allen Hatcher) is a Cambridge Book that focusses more on the algebraic part. Everything basic I said about the second book (the Cambridge Book "Geometry and Topology") applies here as well: it has exercises (without solutions), illustrations, examples and proofs. It is harder to read than "Geometry and Topology" and goes deeper. Maybe you start with the "Geometry and Topology" and continue with this one.

*Introduction to Geometry and Topology (Werner Ballmann) is a very compact, structured but still very illustrative book (Birkhäuser). I recommend it as a companion reference book. If you want to quickly look up and understand a definition, you will find it there. The good thing is that the structure of the book and the definitions are self-contained and you don't have to google on the side, look up on wiki and so on. This is not always a matter of course for compact reference books.

