Matches for: Author/Related=van Mill AND Title="Infinite-dimensional topology"

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90a:57025 57N20 (54C55)
van Mill, J.(NL-VUAM)
Infinite-dimensional topology.
Prerequisites and introduction. North-Holland Mathematical Library, 43.
North-Holland Publishing Co., Amsterdam, 1989. xii+401 pp. \$73.25. ISBN 0-444-87133-0

References: 0

Reference Citations: 5

Review Citations: 6

The first six chapters of the book present a thorough introduction to the background materials necessary for an extensive excursion into the topology of infinite-dimensional manifolds modeled on the Hilbert cube. The final two chapters contain an exposition, as well as applications, of the topological characterization of Hilbert cube manifolds due to H. Torunczyk. The book assumes only a familiarity with basic topological concepts (e.g., metric spaces, complete metric spaces, compactness, connectedness, continuity, product spaces). Chapter 2 is titled "Elementary plane topology" and includes the Brouwer fixed point theorem for $I\sp 2$ and the Jordan curve theorem. Chapter 3 is titled "Elementary combinatorial techniques" and contains an introduction to the combinatorial structure of Euclidean spaces, including "Sperner's lemma" and, in turn, the Brouwer fixed point theorem for $I\sp n.$ Chapter 4 is an introduction to dimension theory, Chapter 5 is an introduction to the theory of absolute neighborhood retracts, and Chapter 6 is an introduction to infinite-dimensional manifolds modeled on the Hilbert cube and Hilbert space.

The book functions well as a text. Its proofs are self-contained and there are a reasonable number of exercises. It also serves as a background source as each chapter ends with brief references to original sources.

The author accurately states that "the first part of this book is intended as a text for graduate courses in topology". While a thorough introduction to infinite-dimensional topology is not unexpected, given the title, the materials on dimension theory and ANR theory are remarkably extensive.

Reviewed by John J. Walsh