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Universal spaces and mappings. (English) Zbl 1072.54001

North-Holland Mathematics Studies 198. Amsterdam: Elsevier (ISBN 0-444-51586-0/hbk). xvi, 559 p. (2005).
This book represents an impressive effort by the author to give a unified treatment to the study and construction of universal spaces and even mappings. Of course a space \(T\) (to use the author’s notation) is universal for a class of spaces if \(T\) is a member of the class and each space from the class embeds into \(T\). Certainly a well-known example of a class that has a universal element is the class of compact metric spaces. A more interesting, and representative, example is the class of all normal spaces of a given weight and a fixed finite covering dimension. This example (the construction of the universal) illustrates the key feature in classical ad hoc contructions which is the use of a (needed) factorization theorem. The author overcomes this limitation by a very general construction of a “Containing Space” and the introduction of the new and useful notion of a saturated class. The author is presenting the study of universal spaces and saturated classes as a natural and attractive focus for a broader research program. The initial challenge for most readers will be the need to familiarize themselves with the extensive notation and new concepts. Although these are quite elementary and natural, this is indeed the cost of the sought after generality. The final chapter of the book should provide the successful reader with an excellent launching point for further work in the subject.
The book consists of 10 Chapters. 1: The Construction of Containing Spaces; 2: Saturated Classes; 3: Dimension-like functions; 4: Saturated Classes of spaces with Structure; 5: Completely regular and compact spaces; 6: Mappings and universality; 7: Actions of groups; 8: Containing Spaces and factorizing \(\tau\)-Spectra; 9: Isomotries and Universality; and 10: Concluding remarks and open problems. The author has also taken great care to compile a very extensive and valuable Bibliography.

MSC:

54-02 Research exposition (monographs, survey articles) pertaining to general topology
54B99 Basic constructions in general topology
54C25 Embedding
54D80 Special constructions of topological spaces (spaces of ultrafilters, etc.)
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