Theory of dimensions, finite and infinite.

*(English)*Zbl 0872.54002
Sigma Series in Pure Mathematics. 10. Lemgo: Heldermann. viii, 401 p. (1995).

This book is a revised and enlarged version of the author’s previous book [Dimension theory (1978; Zbl 0401.54029)]. The new title adds three chapters to the four in the earlier version. These new ones contain a large amount of information about infinite-dimensional spaces much of which came to light during the period between the two publications.

Dimension theory is that area of topology which creates a method of uniquely assigning to each topological space \(X\) (in a reasonable class of spaces) an element of \(\{-1, 0, 1, \dots\}\cup\{\infty\}\) called the dimension of \(X\). It attempts to do this in such a way as to agree with our intuitive concept of dimension. So, for example, the dimension of a finite set is 0, that of a line segment is 1, the dimension of a planar figure containing interior is 2, and the dimension of a “solid” is 3. The number -1 is reserved for the dimension of the empty set (for technical reasons).

The author takes a somewhat historical approach by starting with the dimension theory of separable metrizable spaces. This accounts for approximately 1/3 of the book in number of pages. It is inspired by and runs very much in parallel with the famous book [Dimension theory, Princeton Mathematical Series, 4 (1941; Zbl 0060.39808)] by W. Hurewicz and H. Wallman. This chapter sets the stage for the remainder of the book in the sense that dimension theory for larger classes of spaces is expected either to obey the same principles as those for separable metrizable spaces, or to exhibit reasons why they do not do so. It begins with the definition of small inductive dimension, but later defines large inductive dimension and covering dimension. All three definitions agree in the class of separable metrizable spaces. The author points out that, upon leaving the class of separable metrizable spaces, the concept of small inductive dimension has little application.

The second chapter deals with dimension in normal spaces using the large inductive dimension as the vehicle of study. Covering dimension for the class of normal spaces is the focus of chapter 3. Chapter 4 takes on the dimension theory of arbitrary metrizable spaces. In this class, the concepts of covering dimension and large inductive dimension agree, but, unfortunately they do not agree with small inductive dimension. Since the former two yield the most useful theory, the latter is abandoned for this class of spaces.

Although the topic of infinite dimensions is taken up in the previously mentioned book of Hurewicz and Wallman, so many new results have appeared in between that much of that part of their work has been eclipsed. Chapter 5 contains the basic theory of countable-dimensional spaces, those which can be written as the countable union of 0-dimensional subspaces. Such spaces \(X\) can have dimension equal to \(\infty\). Other classes of infinite-dimensional spaces, weakly infinite-dimensional and strongly infinite-dimensional are taken up in Chapter 6.

The final chapter in the book deals with transfinite dimension, both small and large. A space need not have such a dimension at all. For example, in the class of separable metrizable spaces, those which have small inductive dimension must be countable dimensional. Thus, the Hilbert cube turns out to be a compact metrizable space which does not admit a small transfinite dimension. Similarly, for arbitrary metrizable spaces, those which admit large transfinite dimension must be countable dimensional.

As in other volumes which this author has written, the work is frequently interspersed with a great deal of historical background. Due to the overwhelming size of the literature in the field, it would not have been possible or reasonable to include every possible facet of all the events which have preceded the various new results which have come up along the way. The reader should hence be willing to delve a little more into any particular area of this subject which would become interesting, using the author’s historical perspectives as a jumping-off point.

The work comes complete with problems, so that it could be used as a textbook in a graduate course or seminar. The reviewer, nevertheless, sees this more as a reference book with the potential to be used as the launching vehicle for a student or other non-expert trying to learn the area from the ground up. Anybody who works through this wonderful dimension theory book will not only learn some dimension theory (and want to go more deeply into the subject), but also will learn a storehouse full of the basic tools used in all of topology.

Dimension theory is that area of topology which creates a method of uniquely assigning to each topological space \(X\) (in a reasonable class of spaces) an element of \(\{-1, 0, 1, \dots\}\cup\{\infty\}\) called the dimension of \(X\). It attempts to do this in such a way as to agree with our intuitive concept of dimension. So, for example, the dimension of a finite set is 0, that of a line segment is 1, the dimension of a planar figure containing interior is 2, and the dimension of a “solid” is 3. The number -1 is reserved for the dimension of the empty set (for technical reasons).

The author takes a somewhat historical approach by starting with the dimension theory of separable metrizable spaces. This accounts for approximately 1/3 of the book in number of pages. It is inspired by and runs very much in parallel with the famous book [Dimension theory, Princeton Mathematical Series, 4 (1941; Zbl 0060.39808)] by W. Hurewicz and H. Wallman. This chapter sets the stage for the remainder of the book in the sense that dimension theory for larger classes of spaces is expected either to obey the same principles as those for separable metrizable spaces, or to exhibit reasons why they do not do so. It begins with the definition of small inductive dimension, but later defines large inductive dimension and covering dimension. All three definitions agree in the class of separable metrizable spaces. The author points out that, upon leaving the class of separable metrizable spaces, the concept of small inductive dimension has little application.

The second chapter deals with dimension in normal spaces using the large inductive dimension as the vehicle of study. Covering dimension for the class of normal spaces is the focus of chapter 3. Chapter 4 takes on the dimension theory of arbitrary metrizable spaces. In this class, the concepts of covering dimension and large inductive dimension agree, but, unfortunately they do not agree with small inductive dimension. Since the former two yield the most useful theory, the latter is abandoned for this class of spaces.

Although the topic of infinite dimensions is taken up in the previously mentioned book of Hurewicz and Wallman, so many new results have appeared in between that much of that part of their work has been eclipsed. Chapter 5 contains the basic theory of countable-dimensional spaces, those which can be written as the countable union of 0-dimensional subspaces. Such spaces \(X\) can have dimension equal to \(\infty\). Other classes of infinite-dimensional spaces, weakly infinite-dimensional and strongly infinite-dimensional are taken up in Chapter 6.

The final chapter in the book deals with transfinite dimension, both small and large. A space need not have such a dimension at all. For example, in the class of separable metrizable spaces, those which have small inductive dimension must be countable dimensional. Thus, the Hilbert cube turns out to be a compact metrizable space which does not admit a small transfinite dimension. Similarly, for arbitrary metrizable spaces, those which admit large transfinite dimension must be countable dimensional.

As in other volumes which this author has written, the work is frequently interspersed with a great deal of historical background. Due to the overwhelming size of the literature in the field, it would not have been possible or reasonable to include every possible facet of all the events which have preceded the various new results which have come up along the way. The reader should hence be willing to delve a little more into any particular area of this subject which would become interesting, using the author’s historical perspectives as a jumping-off point.

The work comes complete with problems, so that it could be used as a textbook in a graduate course or seminar. The reviewer, nevertheless, sees this more as a reference book with the potential to be used as the launching vehicle for a student or other non-expert trying to learn the area from the ground up. Anybody who works through this wonderful dimension theory book will not only learn some dimension theory (and want to go more deeply into the subject), but also will learn a storehouse full of the basic tools used in all of topology.

Reviewer: L.R.Rubin (Norman)

##### MSC:

54-02 | Research exposition (monographs, survey articles) pertaining to general topology |

54F45 | Dimension theory in general topology |

55M10 | Dimension theory in algebraic topology |