Reviewed work: J. M. Aarts and T. Nishiura. Dimension and extensions. North-Holland Math. Library, Amsterdam, London, New York, and Tokyo, 1993, xii + 331 pp., $106.50. ISBN 0444897402

Henriksen, Melvin

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Claremont CollegesScholarship @ ClaremontAll HMC Faculty Publications and Research HMC Faculty Scholarship1-1-1994Review: J.M. Aarts and T. Nishiura, Dimension andExtensions (Amsterdam, London, New York, andTokyo, 1993)Melvin HenriksenHarvey Mudd CollegeThis Book Review is brought to you for free and open access by the HMC Faculty Scholarship at Scholarship @ Claremont. It has been accepted forinclusion in All HMC Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, pleasecontact scholarship@cuc.claremont.edu.Recommended CitationHenriksen, Melvin. Review of Dimension and Extensions, by J.M. Aarts and T. Nishiura. Bulletin of the American MathematicalSociety 30 (1994): 282–284. DOI: 10.1090/S0273-0979-1994-00464-1APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 30, Number 2, April 1994, Pages 282-284BOOK REVIEWDimension and extensions, by J. M. Aarts and T. Nishiura. North-Holland Math.Library, Amsterdam, London, New York, and Tokyo, 1993, xii + 331 pp.,$106.50. ISBN 0 444 89740 2Johann de Groot would have loved this book! In his 1942 doctoral dissertationhe considered the problem of compactifying a (separable metrizable) space X byadding a metrizable remainder of minimal dimension. This latter dimension is calledthe compactness deciency, def X, of the space X. He formulated a conjecture onhow to characterize this number internally that captured the imagination of scoresof mathematicians. In 1980 Roman Pol disproved this conjecture but only after anew branch of dimension theory had been developed and had taken on a life of itsown. This excellently written, exciting book is a portrait of a living and dynamicarea as well as a monument to de Groot built by his former student, Jan Aarts,and his able coauthor, Togo Nishiura. It should be required reading for anyoneinterested in dimension theory. Regrettably, its high price will prevent it frombeing on as many bookshelves as it should.The rst chapter sets the stage for the rest of the drama of the book. It concen-trates on spaces that are separable and metrizable where the small inductive, thelarge inductive, and the classical Lebesgue covering dimensions coincide. For sucha space X, dene cmpX inductively as follows: (i) cmpX = −1 if X is compact;(ii) for any nonnegative integer n, cmp  n if for each point p and each closed setG not containing p there is a partition S between p and G with cmpS  n − 1;(iii) cmpX = n if cmpX  n and cmpX  n − 1; and (iv) cmpX = 1 if theinequality cmpX  n fails for each n  0. (This would be the denition of indXif \X is compact" in (i) were replaced by \X = ?".) In his thesis de Groot showedthat def X = cmpX (for X separable and metrizable) if n = −1 or 0, and heconjectured that this equality held for every positive integer. The correspondinganalog of the large inductive dimension is denoted by Cmp. Many other dimensionfunctions were introduced in an attempt to split the de Groot problem into moretractable pieces, and it was less than obvious that one was more pertinent than theother. In 1960 Sklyarenko introduced an integer-valued function that associateswith separable metrizable X an integer the authors denote by SklX as follows:SklX  n if X has a base B = fBi : i = 1; 2; : : : g such that for any n+ 1 dierentindices i0; : : : ; in, the intersection of the boundaries of the Uij for 0  j  n iscompact.Inductive denitions are not easy to work with, and it took until 1982 for Polto produce an example of a separable metrizable space X with cmpX = 1 andc©1994 American Mathematical Society0273-0979/94 $1.00 + $.25 per page12 BOOK REVIEWCmpX = def X = 2. Later, Kimura and then Hart showed that the gap betweendef X and cmpX could be made arbitrarily large. In 1988 Kimura was able toshow that def X = SklX. While this may appear to put the nishing touch tothe de Groot problem, it is far from the end of the story|even for the separablemetrizable case. Compactifying is not the only way to extend a space. For example,the corresponding questions for completions are investigated. Covering dimensionis also generalized. The many unsolved problems in this chapter demonstrate howalive this subject remains even in the separable metrizable case.While the material described above occupies less than a quarter of the book, itsets the tone for the rest of it. Once one lifts the restriction to separable metrizablespaces, the classical dimension functions part company, as do their generalizationsin the spirit of de Groot. The authors consider generalizations of dimensions moduloclasses of topological spaces. A unied approach to the various compacticationsthat arise in this kind of dimension theory is developed. A lot of new materialis presented, and unsolved problems abound. To give the rest of the book theattention it deserves would require an encyclopedic review and would embarrassthe reviewer if he failed to convey the excitement one gets by reading this volume.Many topics considered are motivated and placed in proper historical perspective.Generality appears naturally, not just for its own sake. While a prior knowledge ofclassical dimension theory is helpful, the book is largely self-contained except forreferences to older books for some proofs of theorems. The exposition is masterful,and it is clear that the authors are more interested in communication than indisplaying their (considerable) erudition.Melvin HenriksenHarvey Mudd CollegeE-mail address : HENRIKSEN@SIF.CLAREMONT.EDU

Review: J.M. Aarts and T. Nishiura, Dimension and Extensions (Amsterdam, London, New York, and Tokyo, 1993)

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Review: J.M. Aarts and T. Nishiura, Dimension and Extensions (Amsterdam, London, New York, and Tokyo, 1993)

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