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For the past few years I have sporadically been exploring Topology from a number of textbooks which have come my way. In particular, and particularly at the moment, I have been using Steen and Seebach's "Counterexamples in Topology", the 1995 Dover reprint of the 2nd Edition from 1978.

Now I have heard a lot of bad news about this book: a lot of feedback I've had suggests that somehow it's substandard, misleading, and just generally unworthy of being a textbook. But look, it categorises in dictionary format over a hundred distinct topological spaces (as 148 examples, of which there are in some cases multiple subexamples), taking on a wide swathe of point-set topology.

Criticisms I've had include:

a) Point-set topology is a dead-end waste of time.

b) It's riddled with mistakes (yes okay, I'm up to speed on that, I have found a number of these through careful and close study, but then so do many textbooks have a lot of mistakes, and besides, finding them and correcting them is a valid learning experience).

c) The terminology and notation are laughably old-fashioned, and that renders the book dangerously misleading.

d) It's nowhere near complete, there are all sorts of recent developments which haven't been covered.

Okay, so given that the above prove that it should be consigned to the scrap-heap, what other single work contains such a colossal wealth of study material in such a compact form (less than 250 pages and that includes a substantial index and bibliography).

I have been instructed to get Engelking, which has indeed gone on my booklist, and it has also been suggested that I get Hocking and Young (I had that once but found it difficult to read, consisting of a solid wall of sesquipedalian text, and it fell out of my luggage somewhere between Florida and London and I never replaced it). I also have Hausdorff, Kelley, Mendelson, Blackett, Barr, McCarty, Kasriel, Gamelin & Greene, Sutherland and maybe one or two others I can't immediately bring to hand right now. None of them is anywhere near as "fun" as Steen and Seebach, and none is as exhaustive in the material covered.

So, TL;DR: why is Steen and Seebach so unpopular?

Prime Mover
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    As of now I was under the impression that Steen & Seebach was one of the (if not the) standard reference for general topology i.e. about as far from "unpopular" as it gets. Also, Steen & Seebach is a reference, not a textbook – G. Chiusole May 27 '20 at 10:43
  • I agree with @G. Chiusole. For me A&S is simply a reference to look something up, although I rarely use it because usually when I'm working on something (e.g. 1 2 3 4) I already know of better places for the topics I'm interested in (and there's google $\ldots)$ (continued) – Dave L. Renfro May 27 '20 at 10:59
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    I think the last time I even looked at A&S was for this answer. I am a little confused about "The terminology and notation are laughably old-fashioned" however, as I just looked through it and I don't see anything of concern with notation. It uses the current union and intersection symbols, not $+$ and $-$ signs (in fact, the $+$ and $-$ sign usage mostly ended after the 1940s), the current interval endpoint conventions are followed (i.e. $[$ for including the endpoint and $($ for not including the endpoint), and other things. (continued) – Dave L. Renfro May 27 '20 at 11:05
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    As for terminology, I guess I'm not all that current on what THE current terms are for many of the notions, so I'm probably not a good judge there. Nonetheless, based on your comment I did check one term --- spaces satisfying the finite covering property are called "compact", not "bicompact". But I think the use of "bicompact" mostly fell out of favor by the early to mid 1950s, with Kelley's General Topology possibly being a major reason (just a guess). – Dave L. Renfro May 27 '20 at 11:06
  • @Dave L. Renfro The definitions for separation axioms have changed sense, e.g. S&S say a Regular space is one which is T3 and T1, while other sources (eg. Wikipedia) have a T3 space as one which is both Regular and T1, defining Regular as what S&S have as T3 and T3 for what S&S has as Regular. Same with T4, T5, etc. And they say "hyperconnected" rather than "irreducible" and so on. As for using it as a text book, I am working through it and proving all the assertions made, for want of a better approach to studying topology. – Prime Mover May 27 '20 at 11:11
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    I'm not sure about whether "hyperconnected" is a good example, but the separation axioms (beginning with $T_3)$ are well known to vary by author, mainly as to whether $T_1$ is included in the definition of regular and normal. As for Wikipedia, I would not use that for anything except an overview and pointers to the literature, as what's written is highly dependent on the background of the writers (e.g. this entry is misleadingly incomplete, and see my and Thomson's comments here about Froda's theorem). – Dave L. Renfro May 27 '20 at 11:23
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    Incidentally, given what you've said, for self-study I recommend Willard's General Topology and/or Wilansky's Topology for Analysis, each of which has relatively recently been available as cheap Dover reprints and each of which has a strong focus on counterexamples (especially Wilansky's book; however, Wilansky's book has very little on connectedness, continua, and the like). – Dave L. Renfro May 27 '20 at 11:38
  • I've very probably been guilty of taking seriously the claims of people who perhaps are somewhat partisan about the books they like. For definite, on the subject of topology people are very opinionated about those books which they think are gospel and those which they think are rubbish. I'll put the Willard and the Wilansky on my to-get list. Won't be yet because I haven't been paid properly for a few months. – Prime Mover May 27 '20 at 14:35
  • Steen & Seebach is a lousy text, but it was never intended to be one; it is a useful collection of counterexamples. Its terminology is within the range of current usage (including the bizarre but common failure to recognize that the $T_i$ separation axioms should form a genuine hierarchy). Some of its examples could be replaced with better ones, some of the descriptions could be improved, and some interesting topics that have grown out of set-theoretic topology are of course missing, but I don’t know of any better replacement. I also recommend Willard’s General Topology. – Brian M. Scott May 27 '20 at 18:47
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    Oh, and I used Hocking & Young in a second undergraduate topology course back around 1966 and quite disliked it; I can’t say, however, how much of that dislike was due to the book and how much to the fact that I simply wasn’t (and am not) nearly as much interested in continua theory and algebraic topology as in point-set and set-theoretic topology. – Brian M. Scott May 27 '20 at 18:52

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