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E-grāmata: Infinite-Dimensional Topology: Prerequisites and Introduction

(Vrije Universiteit, Department of Mathematics and Computer Science, Amsterdam, The Netherlands)
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Infinite-Dimensional Topology: Prerequisites and IntroductionPalielināt
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The first part of this book is a text for graduate courses in topology. In chapters 1 - 5, part of the basic material of plane topology, combinatorial topology, dimension theory and ANR theory is presented. For a student who will go on in geometric or algebraic topology this material is a prerequisite for later work. Chapter 6 is an introduction to infinite-dimensional topology; it uses for the most part geometric methods, and gets to spectacular results fairly quickly. The second part of this book, chapters 7 & 8, is part of geometric topology and is meant for the more advanced mathematician interested in manifolds.

The text is self-contained for readers with a modest knowledge of general topology and linear algebra; the necessary background material is collected in chapter 1, or developed as needed.

One can look upon this book as a complete and self-contained proof of Toruńczyk's Hilbert cube manifold characterization theorem: a compact ANR X is a manifold modeled on the Hilbert cube if and only if X satisfies the disjoint-cells property. In the process of proving this result several interesting and useful detours are made.

Recenzijas

"...recommended to anyone who wishes to get familiar with infinite-dimensional topology and at the same time learn about some its most beautiful results." --Zentralblatt für Mathematik

1. Extension Theorems. Topological Spaces. Linear Spaces. Function
Spaces. The Michael Selection Theorem and Applications. AR's and ANR's. The
Borsuk Homotopy Extension Theorem.2. Elementary Plane Topology. The Brouwer
Fixed-Point Theorem and Applications. The Borsuk-Ulam Theorem. The Poincaré
Theorem. The Jordan Curve Theorem.3. Elementary Combinatorial Techniques.
Affine Notions. Simplexes. Triangulation. Simplexes in Rn. The Brouwer
Fixed-Point Theorem. Topologizing a Simplical Complex.4. Elementary Dimension
Theory. The Covering Dimension. Zero-Dimensional Spaces. Translation into
Open Covers. The Imbedding Theorem. The Inductive Dimension Functions ind and
Ind. Mappings into Spheres. Totally Disconnected Spaces. Various Kinds of
Infinite-Dimensionality.5. Elementary ANR Theory. Some Properties of ANR's. A
Characterization of ANR's and AR's. Hyperspaces and the AR-Property. Open
Subspaces of ANR's. Characterization of Finite-Dimensional ANR's and AR's.
Adjunction Spaces of Compact A(N)R's.6. An Introduction to
Infinite-Dimensional Topology. Constructing New Homeomorphisms from Old.
Z-Sets. The Estimated Homeomorphism Extension Theorem for Compacta in s. The
Estimated Homeomorphism Extension Theorem. Absorbers. Hilbert Space is
Homeomorphic to the Countable Infinite Product of Lines. Inverse Limits.
Hilbert Cube Factors.7. Cell-Like Maps and Q-Manifolds. Cell-Like Maps and
Fine Homotopy Equivalences. Z-Sets in ANR's. The Disjoint-Cells Property.
Z-Sets in Q-Manifolds. Toruńczyk's Approximation Theorem and
Applications. Cell-Like Maps. The Characterization Theorem.8. Applications.
Infinite Products. Keller's Theorem. Cone Characterization of the Hilbert
Cube. The Curtis-Schori-West Hyperspace Theorem.What Next? Bibliography.
Subject Index.
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