Dimension Theory: A Selection of Theorems and Counterexamples

Dimension.  It’s a concept we think we understand most of the time.  But a proof that R^n and R^m are topologically equivalent if and only if n = m didn’t occur until 1911.  This struggle was the beginning of the field of dimension theory, which is the study of certain functions from topological spaces into the extended integers (ie, the integers, union with \( \infty \)). 

 

Charalambous’ book sets out to collect some results on the topic of dimension theory, which are scattered in the literature, although it does not aim to be exhaustive in this collection. Between the numerous citations and the large bibliography, results are well-documented here.  Many chapters begin with a short paragraph containing several references.  

 

The author points out that this book could serve as the text for a graduate course. I agree. Anyone who has studied the usual separation properties (\( T_{1} \), \( T_{2} \), etc.) in a basic point-set topology course is in a position to engage with this text.   However, the chapters are short and dense and are best suited to the very motivated.  In addition, the index is minimal; it requires effort to find definitions of some terms introduced in earlier chapters.   Exercises are included at the end of each chapter, and the extensive bibliography should be helpful particularly to those who become enamored with the subject.

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Michele Intermont is an Associate Professor of Mathematics at Kalamazoo College.  Her specialty is algebraic topology.