Geometric aspects of general topology [electronic resource] /
Sakai, Katsuro.author.
This book is designed for graduate students to acquire knowledge of dimension theory, ANR theory (theory of retracts), and related topics. These two theories are connected with various fields in geometric topology and in general topology as well. Hence, for students who wish to research subjects in general and geometric topology, understanding these theories will be valuable. Many proofs are illustrated by figures or diagrams, making it easier to understand the ideas of those proofs. Although exercises as such are not included, some results are given with only a sketch of their proofs. Completing the proofs in detail provides good exercise and training for graduate students and will be useful in graduate classes or seminars. Researchers should also find this book very helpful, because it contains many subjects that are not presented in usual textbooks, e.g., dim X × I = dim X + 1 for a metrizable space X; the difference between the small and large inductive dimensions; a hereditarily infinite-dimensional space; the ANR-ness of locally contractible countable-dimensional metrizable spaces; an infinite-dimensional space with finite cohomological dimension; a dimension raising cell-like map; and a non-AR metric linear space. The final chapter enables students to understand how deeply related the two theories are. Simplicial complexes are very useful in topology and are indispensable for studying the theories of both dimension and ANRs. There are many textbooks from which some knowledge of these subjects can be obtained, but no textbook discusses non-locally finite simplicial complexes in detail. So, when we encounter them, we have to refer to the original papers. For instance, J.H.C. Whitehead's theorem on small subdivisions is very important, but its proof cannot be found in any textbook. The homotopy type of simplicial complexes is discussed in textbooks on algebraic topology using CW complexes, but geometrical arguments using simplicial complexes are rather easy.
Description based upon print version of record.
""Preface""; ""Contents""; ""Chapter 1: Preliminaries""; ""1.1 Terminology and Notation""; ""1.2 Banach Spaces in the Product of Real Lines""; ""Notes for Chap. 1""; ""References""; ""Chapter 2: Metrization and Paracompact Spaces""; ""2.1 Products of Compact Spaces and Perfect Maps""; ""2.2 The Tietze Extension Theorem and Normalities""; ""2.3 Stone's Theorem and Metrization""; ""2.4 Sequences of Open Covers and Metrization""; ""2.5 Complete Metrizability""; ""2.6 Paracompactness and Local Properties""; ""2.7 Partitions of Unity""; ""2.8 The Direct Limits of Towers of Spaces""
""2.9 The Limitation Topology for Spaces of Maps""""2.10 Counter-Examples""; ""Notes for Chap.2""; ""References""; ""Chapter 3: Topology of Linear Spaces and Convex Sets""; ""3.1 Flats and Affine Functions""; ""3.2 Convex Sets""; ""3.3 The Hahnâ€?Banach Extension Theorem""; ""3.4 Topological Linear Spaces""; ""3.5 Finite-Dimensionality""; ""3.6 Metrizability and Normability""; ""3.7 The Closed Graph and Open Mapping Theorems""; ""3.8 Continuous Selections""; ""3.9 Free Topological Linear Spaces""; ""Notes for Chap.3""; ""References""; ""Chapter 4: Simplicial Complexes and Polyhedra""
""4.1 Simplexes and Cells""""4.2 Complexes and Subdivisions""; ""4.3 Product Complexes and Homotopy Extension""; ""4.4 PL Maps and Simplicial Maps""; ""4.5 The Metric Topology of Polyhedra""; ""4.6 Derived and Barycentric Subdivisions""; ""4.7 Small Subdivisions""; ""4.8 Admissible Subdivisions""; ""4.9 The Nerves of Open Covers""; ""4.10 The Inverse Limits of Metric Polyhedra""; ""4.11 The Mapping Cylinders""; ""4.12 The Homotopy Type of Simplicial Complexes""; ""4.13 Weak Homotopy Equivalences""; ""4.14 Appendix: Homotopy Groups""; ""References""; ""Chapter 5: Dimensions of Spaces""
""Chapter 6: Retracts and Extensors""""6.1 The Dugundji Extension Theorem and ANEs""; ""6.2 Embeddings of Metric Spaces and ANRs""; ""6.3 Small Homotopies and LEC Spaces""; ""6.4 The Homotopy Extension Property""; ""6.5 Complementary Pairs of ANRs""; ""6.6 Realizations of Simplicial Complexes""; ""6.7 Fine Homotopy Equivalences""; ""6.8 Completions of ANRs and Uniform ANRs""; ""6.9 Homotopy Types of Open Sets in ANRs""; ""6.10 Countable-Dimensional ANRs""; ""6.11 The Local n-Connectedness""; ""6.12 Finite-Dimensional ANRs""; ""6.13 Embeddings into Finite-Dimensional ARs""; ""References""
""Chapter 7: Cell-Like Maps and Related Topics""
Includes bibliographical references and index.
This book is designed for graduate students to acquire knowledge of dimension theory, ANR theory (theory of retracts), and related topics. These two theories are connected with various fields in geometric topology and in general topology as well. Hence, for students who wish to research subjects in general and geometric topology, understanding these theories will be valuable. Many proofs are illustrated by figures or diagrams, making it easier to understand the ideas of those proofs. Although exercises as such are not included, some results are given with only a sketch of their proofs. Completing the proofs in detail provides good exercise and training for graduate students and will be useful in graduate classes or seminars. Researchers should also find this book very helpful, because it contains many subjects that are not presented in usual textbooks, e.g., dim X × I = dim X + 1 for a metrizable space X; the difference between the small and large inductive dimensions; a hereditarily infinite-dimensional space; the ANR-ness of locally contractible countable-dimensional metrizable spaces; an infinite-dimensional space with finite cohomological dimension; a dimension raising cell-like map; and a non-AR metric linear space. The final chapter enables students to understand how deeply related the two theories are. Simplicial complexes are very useful in topology and are indispensable for studying the theories of both dimension and ANRs. There are many textbooks from which some knowledge of these subjects can be obtained, but no textbook discusses non-locally finite simplicial complexes in detail. So, when we encounter them, we have to refer to the original papers. For instance, J.H.C. Whitehead's theorem on small subdivisions is very important, but its proof cannot be found in any textbook. The homotopy type of simplicial complexes is discussed in textbooks on algebraic topology using CW complexes, but geometrical arguments using simplicial complexes are rather easy.
Topology.
Convex geometry .
Discrete geometry.
Geometry.
Functional analysis.
Topology.
Convex and Discrete Geometry.
Geometry.
Functional Analysis.