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Though there are several posts discussing the reference books for topology, for example best book for topology. But as far as I looked up to, all of them are for the purpose of learning topology or rather on introductory level.

I am wondering if there is a book or a set of books on topology like Rudin's analysis books, S.Lang's algebra, Halmos' measure (or to be more updated, Bogachev's measure theory), etc, serving as standard references.

Probably, such a book should hold characteristics such as being self-contained, covering the most of classical results, and other good properties you can name.

In the end, I only have the interest on general-topology (topological space, metrization, compactification...), and optionally differential topology (manifolds). So please don't divert into algebraic context.

Cheers.

---------------------update-----------------------

Thanks for all the delightful replies. I will try to quickly check the books mentioned beneath. And I may accept the answer which is most closed to my personal flavor.

Sorry for others. It's a pity no multiple acceptance can be made for such a ref-request question.

Greetings.

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11 Answers 11

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For general topology, it is hard to beat Ryszard Engelking's "General Topology". It starts at the very basics, but goes through quite advanced topics. It may be perhaps a bit dated, but it is still the standard reference in general topology.

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    $\begingroup$ Dammit! I was about to hit the "Post" button with that answer! :-) $\endgroup$ – Asaf Karagila♦ Mar 9 '12 at 11:09
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    $\begingroup$ Now add to that that a true great mind will deny its greatness, and we go circular: If I'm calling you a great mind, I'm calling myself a great mind, thus proving that I am in fact not a great mind and therefore falsifying the fact that either you are a great mind, or that we think alike. However we do think alike so you're not a great mind either. From this we can deduce that if I do not agree that you're a great mind you're saying that I have a great mind and by denying I'm accepting this premise and therefore you have a great mind, which then reduces us to the previous case! $\endgroup$ – Asaf Karagila♦ Mar 9 '12 at 11:13
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    $\begingroup$ This is to add another vote for Engelking's book. I've had a copy since 1977 and I have found it to be the single best reference for post-Munkres level general topology. It is also one of the very few texts (in fact, the only text I can think of right now) that makes use of cardinal functions. I also wish to second Greinecker's suggestion of Willard's text (I got Willard's book in 1976 and have covered it front to back in 3 semesters, one of which was a directed reading), which is pretty much the standard introductory text for those who intend to continue further in general topology. $\endgroup$ – Dave L. Renfro Mar 9 '12 at 15:22
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    $\begingroup$ Uh-you guys ARE aware that there's a second edition of Engelking dating from 1989 that's vastly expanded and updated,right? The problem is it costs an arm and a leg and it can only be obtained from Heldermann-Verlag. If you can locate a copy of the second edition,I STRONGLY advise you to get a copy. It truly is one of the world's great mathematics textbooks. $\endgroup$ – Mathemagician1234 Apr 27 '12 at 15:14
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    $\begingroup$ @Mathemagician1234 considering that the OP's link is to the second edition, I suspect that at least a few of the participants in this thread is aware of that. :) $\endgroup$ – Willie Wong May 22 '12 at 12:27
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Stephen Willard, General Topology

This book is less complete than Engelking, but still contains enough material to make a good reference book. It is also quite cheap, as a Dover book.

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  • $\begingroup$ +1. If you can't get a hold of the second edition of Engelking, this really is your best choice. $\endgroup$ – Mathemagician1234 Apr 27 '12 at 15:15
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The following 3 volume set (translated from Russian, edited by A. V. Arhangelskii) deserves to be mentioned among references for general topology, too. It is part of Encyclopaedia of Mathematical Sciences series.

  • General topology. I. Basic concepts and constructions. Dimension theory. Encyclopaedia of Mathematical Sciences, 17. Springer-Verlag, Berlin, 1990. ISBN 3-540-18178-4 Google Books link, DOI:10.1007/978-3-642-61265-7_1, MR1077251

  • General topology. II. Compactness, homologies of general spaces. Encyclopaedia of Mathematical Sciences, 50. Springer-Verlag, Berlin, 1996. ISBN 3-540-54695-2 Google Books link, DOI:10.1007/978-3-642-77030-2, MR1392480

  • General topology. III. Paracompactness. Function spaces. Descriptive theory. Encyclopaedia of Mathematical Sciences, 51. Springer-Verlag, Berlin, 1995. ISBN 3-540-54698-7 Google Books link, DOI:10.1007/978-3-662-07413-8, MR1416131.

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  • $\begingroup$ Hi Martin, I intend to read this collection of books. I just want to know: is this collection really good? Is there any better? I realized that these books do not contain any exercise, so could you point me problem books on topology that can complement those books? $\endgroup$ – user477271 Jan 22 '18 at 15:36
  • $\begingroup$ @rfloc I am not sure whether I can answer your questions, but I would suggest to continue this discussion in general topology chatroom rather than here. $\endgroup$ – Martin Sleziak Jan 22 '18 at 15:38
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The Handbook of Set-Theoretic Topology is a great reference on many advanced areas of general topology.

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  • $\begingroup$ Review of Kenneth Kunen, Jerry E. Vaughan (editors), Handbook of Set-Theoretic Topology from Journal of Symbolic Logic: projecteuclid, jstor $\endgroup$ – Martin Sleziak May 22 '12 at 13:42
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A really nice book about general topology is "Topology" by "James Dugundji". For those who can read portuguese I'd recommend "Elementos de Topologia Geral" by "Elon Lages Lima" - a great book.

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Please look at "Topology and groupoids", http://www.bangor.ac.uk/r.brown/topgpds.html

which is published privately to keep the price down, and an e-version for £5 is available through the above site.

The first part is a geometric account of general topology, with motivation for definitions and theorems, starting with the neighbourhood axioms, as more intuitive, and then proceeding to open sets, etc. There is a gradual introduction to universal properties, so that topologies are often defined in order to be able to construct various kinds of continuous functions. It has a lot on identification spaces, adjunction spaces, finite cell complexes, and also an introduction to a topology on function spaces to give a convenient category of spaces.

The second part is on the use of the fundamental groupoid in algebraic topology, allowing more powerful theorems with simpler proofs.

A review is at

http://mathdl.maa.org/mathDL/19/?pa=reviews&sa=viewBook&bookId=69421

As said elsewhere on this site, it does not cater so well for the needs of analysts, but they also ought to know about universal properties!

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    $\begingroup$ +1. You've written an outstanding book and you're absolutely right:It does play down the analytic aspects of point-set topology. But that's just fine because there are a legion of textbooks that present that material and do it very well. There was a need for a modern presentation that emphasizes universal properties and presents point-set theory in completely modern language and prepares the reader for a serious graduate course in algebraic topology,such as from tom Dieck or May's books.Your book does that wonderfully. $\endgroup$ – Mathemagician1234 Apr 27 '12 at 15:18
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One other old stand-by is J.L. Kelley's General Topology, published as GTM 27. It is quite good especially if you are approaching the topic with the eye of an analyst. (In the preface he professed that he wanted to subtitle the book "What Every Young Analyst Should Know".)

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Another book which might be worth mentioning in this context (although this is different from other books mentioned here; it contains overview of many various areas, but no proofs of the results given there):

Encyclopedia of General Topology; Edited by: Klaas Pieter Hart, Jun-iti Nagata and Jerry E. Vaughan, Elsevier, 2003, ISBN: 978-0-444-50355-8. sciencedirect, Elsevier, MR2049453, Zbl 1059.54001 .

Many of the authors provide the chapters which their contributed to this book freely on their websites.

Quote from the preface:

Thus the book provides a source where the specialist and nonspecialist alike can find short introductions to both the basic theory and the newest developments in General Topology.

Because the book is designed for the reader who wants to get a general view of the terminology with minimal time and effort there are very few proofs given; on occasion a sketch of an argument will be given, more to illustrate a notion than to justify a claim.

A reader who wants to study the subject matter of one or more of the articles systematically (or who wants to see the proof of a particular result) will find sufficient references at the end of each article as well as in the books in our list of standard references.

In connection with this question, the list of references given in the preface might be of interest too. Google Books link.

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  • $\begingroup$ Wow...I have met Dr. Hart personally. Have no idea he had written such a book. $\endgroup$ – newbie May 22 '12 at 13:37
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Kazimier Kuratowski has written a two volume treatise on Topology which focuses more on General Topology. There is also an English translation available. I did not read it entirely, but by reading a few excerpts and looking through the contents it seems to be quite comprehensive.

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  • $\begingroup$ I will just add that there is French, Russian and English version. (The French one being the original.) And maybe some other translation. $\endgroup$ – Martin Sleziak Mar 17 '15 at 6:33
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Bourbaki's General Topology is in my opinion the best reference on General Topology. (The English version has two volumes.)

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Topology without tears of Sidney Morris is a great book to learn topology. It's written in a very attractive way, has a lot exercises and covers a great deal of material in general topology and some material in real analysis. It has been updated recently and now has 12 chapters instead of 10 when is was firstly released. It has also appendices that include Hausdorff dimension, dynamical systems, set theory filters and nets and other topics. I read that it will be updated in the future and will contain 15 chapters and more appendices. You can download for free everywhere but I insist to try finding the recent updated version that I mentioned above. My personal opinion is that this book is like a bible for someone who likes general topology.

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  • $\begingroup$ I do not think this answers the question which asks for reference rather than introductory texts. In fact, in another question which asks about textbooks, Topology without tears is mention in the currently accepted answer. $\endgroup$ – Martin Sleziak Sep 14 '16 at 6:11

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