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This book covers the fundamental results of the dimension theory of metrizable spaces, especially in the separable case. Its distinctive feature is the emphasis on the negative results for more general spaces, presenting a readable account of numerous counterexamples to well-known conjectures that have not been discussed in existing books. Moreover, it includes three new general methods for constructing spaces: Mrowka's psi-spaces, van Douwen's technique of assigning limit points to carefully selected sequences, and Fedorchuk's method of resolutions.
Accessible to readers familiar with the standard facts of general topology, the book is written in a reader-friendly style suitable for self-study. It contains enough material for one or more graduate courses in dimension theory and/or general topology. More than half of the contents do not appear in existing books, making it also a good reference for libraries and researchers.
List of contents
- Topological Spaces. - The Three Main Dimension Functions. - The Countable Sum Theorem for Covering Dimension. - Urysohn Inequalities. - The Dimension of Euclidean Spaces. - Connected Components and Dimension. - Factorization and Compactification Theorems for Separable Metric Spaces. - Coincidence, Product and Decomposition Theorems for Separable Metric Spaces. - Universal Spaces for Separable Metric Spaces of Dimension at Most n. - Axiomatic Characterization of the Dimension of Separable Metric Spaces. - Cozero Sets and Covering Dimension dim0. - -Spaces and the Failure of the Sum and Subset Theorems for dim0. - The Inductive Dimension Ind0. - Two Classical Examples. - The Gap Between the Covering and the Inductive Dimensions of Compact Hausdorff Spaces. - Inverse Limits and N-Compact Spaces. - Some Standard Results Concerning Metric Spaces. - The Mardesi´c Factorization Theorem and the Dimension of Metrizable Spaces. - A Metrizable Space with Unequal Inductive Dimensions. - No Finite Sum Theorem for the Small Inductive Dimension of Metrizable Spaces. - Failure of the Subset Theorem for Hereditarily Normal Spaces. - A Zero-Dimensional, Hereditarily Normal and Lindelöf Space Containing Subspaces of Arbitrarily Large Dimension. - Cosmic Spaces and Dimension. - n-Cardinality and Bernstein Sets. - The van Douwen Technique for Constructing Counterexamples. - No Compactification Theorem for the Small Inductive Dimension of Perfectly Normal Spaces. - Normal Products and Dimension. - Fully Closed and Ring-Like Maps. - Fedorcuk's Resolutions. - Compact Spaces Without Intermediate Dimensions. - More Continua with Distinct Covering and Inductive Dimensions. - The Gaps Between the Dimensions of Normal Hausdorff Spaces.
Report
"The monograph contains a great deal of useful and up-to-date material on dimension theory; the exposition is transparent and well organized which makes it possible to use this work both as a textbook of dimension theory and a base of research projects in numerous areas." (Vladimir Tkachuk, zbMATH 1471.54001, 2021)
