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General topology. Rev. and compl. ed. (English) Zbl 0684.54001
Sigma Series in Pure Mathematics, 6. Berlin: Heldermann Verlag. viii, 529 p. DM 148.00 (1989).
As the title indicates, this is an updated version of the author’s 1977 book General Topology [see the review in Zbl 0373.54002].
Since its appearance that book has become the standard reference in general topology. The reason for this is clear for those who know the work: its main body contains everything a beginning researcher in general topology should know (ideally speaking anyway), the exercises and problems extend and broaden this knowledge considerably and also manage to give an occasional taste of more specialized areas of research, the bibliography is goldmine for those who want to read the originals, and on top of all this the author is an excellent expositor.
The major drawback of the old edition, namely that it was no longer available, is now resolved with this revision. Those who expected a major overhaul will be somewhat disappointed. Indeed, rather than drastically rewriting the book, the author chose to make a thorough update. In his own words: “Important new results related to the topics discussed in the first edition were added, as were some older results which have recently proved to be important.” The bibliography reflects this as well. This new edition deserves the same praise as the first one.
Are there any negative points? Not really, except maybe the fact that a few times the author uses terms that are not common usage, best-known are ‘functionally closed (open) set’ instead of ‘(co)zero-set’ and ‘open (closed) domain’ instead of ‘regular open (closed) set’; also the author does a lot of indexing, like in the definition of compactness: “... for every open cover \(\{U_ S\}_{s\in S}\) of the space X there exists a finite set \(\{s_ 1,s_ 2,...,s_ k\}\subset S\) such that \(X=U_{s_ 1}\cup U_{s_ 2}\cup...\cup U_{s_ k}.''\) This does not really reflect the statement “every open cover has a finite subcover”, but is more a statement about index open covers. This is of course a matter of taste and style and one may like it or not. In no way do these points undermine the usefulness of the book.
I cannot but heartily recommend it to anyone with some interest in General Topology.
Reviewer: K.P.Hart

MSC:
54-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to general topology
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