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Dimension and extensions. (English) Zbl 0873.54037
North-Holland Mathematical Library. 48. Amsterdam: North-Holland. xii, 331 p. (1993).
From the authors’ ‘Preface’: “The problem of de Groot concerned compactifications of spaces by means of an adjunction of a set of minimal dimension. This minimal dimension was called the compactness deficiency of a space. \(\dots\) The compactness degree of de Groot was defined by the replacement of the empty space with compact spaces in the [initial step of the inductive] definition of the small inductive dimension. [De Groot conjectured] that this dimension function would internally characterize the compactness deficiency. \(\dots\) It took almost fifty years to settle the original problem of de Groot. \(\dots\) In 1980 Pol produced an example [a subspace of \(E^4]\) to show that the compactness degree was not the right candidate for characterizing the compactness deficiency. Then Kimura showed in 1988 that the compactness dimension function that was introduced by Sklyarenko in 1960 characterized the compactness deficiency [for separable metric spaces]. [From] 1942 to the present, a substantial theory of the extension of dimension theory to dimension-like functions evolved. This book is a presentation of the current status of [these] problems”.
One can distinguish two streams in the book. The one is the basic material of the inductive theory of dimension. The other is the discussion of the dimension-like functions arising on the base of de Groot’s problem; this is a big material, as the degree can be considered not necessarily modulo compact spaces, and compactifications can be replaced by other kinds of extensions. There is also a part of the book (Chapter I) devoted to presentation of the example of Pol and the theorem of Kimura, which can be considered as the core of the book.
However, the basic material is presented in such a way that neither the beginner nor a specialist will be satisfied. The comments on the early development of dimension theory are both not so deep and not so exhaustive. The “Historical comments” at the ends of the chapters, although containing many details, look simply as lists of names and dates. There are theorems of the basic material which are cited without proofs, for instance, \(\text{ind} E^n =n\). The proofs of some basic theorems used in the reasonings in Chapter I are removed to chapters in the middle of the book.
Concerning the second stream mentioned before, and this is confirmed by the authors in Chapter III, that each theory of dimension has six or seven theorems which characterize the theory. The number of theories is much more higher, it is potentially infinite. So, the Chapters II-VI, devoted to various dimension degrees and various deficiences, consist in discussion of these theorems in enormously many situations. This looks like an attempt of a codification.

54F45 Dimension theory in general topology
54-02 Research exposition (monographs, survey articles) pertaining to general topology
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)