Kuratowski / Sneddon / Stark Introduction to Set Theory and Topology

2. Auflage 2014
ISBN: 978-1-4831-5163-2
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 352 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

Introduction to Set Theory and Topology describes the fundamental concepts of set theory and topology as well as its applicability to analysis, geometry, and other branches of mathematics, including algebra and probability theory. Concepts such as inverse limit, lattice, ideal, filter, commutative diagram, quotient-spaces, completely regular spaces, quasicomponents, and cartesian products of topological spaces are considered. This volume consists of 21 chapters organized into two sections and begins with an introduction to set theory, with emphasis on the propositional calculus and its application to propositions each having one of two logical values, 0 and 1. Operations on sets which are analogous to arithmetic operations are also discussed. The chapters that follow focus on the mapping concept, the power of a set, operations on cardinal numbers, order relations, and well ordering. The section on topology explores metric and topological spaces, continuous mappings, cartesian products, and other spaces such as spaces with a countable base, complete spaces, compact spaces, and connected spaces. The concept of dimension, simplexes and their properties, and cuttings of the plane are also analyzed. This book is intended for students and teachers of mathematics.

Autoren/Hrsg.

Kuratowski, Kazimierz

Sneddon, I. S.

Stark, M.

Inhaltsverzeichnis

1;Front Cover;1
2;Introduction to Set Theory and Topology;4
3;Copyright Page;5
4;Table of Contents;6
5;FOREWORD TO THE FIRST ENGLISH EDITION;12
6;FOREWORD TO THE SECOND ENGLISH EDITION;14
7;Part I: SET THEORY;16
7.1;INTRODUCTION TO PART I;18
7.2;CHAPTER I. PROPOSITIONAL CALCULUS;24
7.2.1;1. The disjunction and conjunction of propositions;24
7.2.2;2. Negation;25
7.2.3;3. Implication;26
7.2.4;Exercises;27
7.3;CHAPTER II. ALGEBRA OF SETS. FINITE OPERATIONS;28
7.3.1;1. Operations on sets;28
7.3.2;2. Inter-relationship with the propositional calculus;29
7.3.3;3. Inclusion;31
7.3.4;4. Space. Complement of a set;33
7.3.5;5. The axiomatics of the algebra of sets;34
7.3.6;6. Boolean algebra.+ Lattices;35
7.3.7;7. Ideals and filters;37
7.3.8;Exercises;37
7.4;CHAPTER III. PROPOSITIONAL FUNCTIONS.CARTESIAN PRODUCTS;40
7.4.1;1. The operation;40
7.4.2;2. Quantifiers;41
7.4.3;3. Ordered pairs;43
7.4.4;4. Cartesian product;43
7.4.5;5. Propositional functions of two variables. Relations;44
7.4.6;6. Cartesian products of n sets. Propositional functions of n variables;47
7.4.7;7. On the axiomatics of set theory;48
7.4.8;Exercises;50
7.5;CHAPTER IV. THE MAPPING CONCEPT. INFINITE OPERATIONS. FAMILIES OF SETS;51
7.5.1;1. The mapping concept;51
7.5.2;2. Set-valued mappings;53
7.5.3;3. The mapping;54
7.5.4;4. Images and inverse images determined by a mapping;55
7.5.5;5· The operations U R and n R. Covers;56
7.5.6;6. Additive and multiplicative families of sets;57
7.5.7;7. Borel families of sets;59
7.5.8;8. Generalized cartesian products;60
7.5.9;Exercises;61
7.6;CHAPTER V. THE CONCEPT OF THE POWER OF A SET. COUNTABLESETS;66
7.6.1;1. One-to-one mappings;66
7.6.2;2. Power of a set;68
7.6.3;3. Countable sets;69
7.6.4;Exercises;72
7.7;CHAPTER VI. OPERATIONS ON CARDINAL NUMBERS. THE NUMBERS a AND c;74
7.7.1;1. Addition and multiplication;74
7.7.2;2. Exponentiation;76
7.7.3;3. Inequalities for cardinal numbers;80
7.7.4;4. Properties of the number c;82
7.7.5;Exercises;85
7.8;CHAPTER VII. ORDER RELATIONS;86
7.8.1;1. Definitions;86
7.8.2;2. Similarity. Order types;86
7.8.3;3. Dense ordering;88
7.8.4;4. Continuous ordering;88
7.8.5;Exercises;91
7.9;CHAPTER VIII. WELL ORDERING;92
7.9.1;1. Well ordering;92
7.9.2;2. Theorem on transfinite induction;93
7.9.3;3. Theorems on the comparison of ordinal numbers;93
7.9.4;4. Sets of ordinal numbers;96
7.9.5;5. The number O;96
7.9.6;6. The arithmetic of ordinal numbers;98
7.9.7;7. The well-ordering theorem;101
7.9.8;8. Definitions by transfinite induction;103
7.9.9;Exercises;106
8;Part II: TOPOLOGY;108
8.1;INTRODUCTION TO PART II;110
8.2;CHAPTER IX. METRIC SPACES. EUCLIDEAN SPACES;116
8.2.1;1. Metric spaces;116
8.2.2;2. Diameter of a set. Bounded spaces. Bounded mappings;117
8.2.3;3. The Hubert cube;118
8.2.4;4. Convergence of a sequence of points;118
8.2.5;5. Properties of the limit;119
8.2.6;6. Limit in the cartesian product;120
8.2.7;7. Uniform convergence;122
8.2.8;Exercises;123
8.3;CHAPTER X. TOPOLOGICAL SPACES;124
8.3.1;1. Definition. Closure axioms;124
8.3.2;2. Relations to metric spaces;124
8.3.3;3. Further algebraic properties of the closure operation;126
8.3.4;4. Closed sets. Open sets;127
8.3.5;5. Operations on closed sets and open sets;128
8.3.6;6. Interior points. Neighbourhoods;130
8.3.7;7. The concept of open set as the primitive term of the notion of topological space;131
8.3.8;8. Base and subbase;132
8.3.9;9. Relativization. Subspaces;133
8.3.10;10. Comparison of topologies;133
8.3.11;11. Cover of a space;134
8.3.12;Exercises;135
8.4;CHAPTER XI. BASIC TOPOLOGICAL CONCEPTS;138
8.4.1;1. Borel sets;138
8.4.2;2. Dense sets and boundary sets;139
8.4.3;3. f1-spaces. f2-spaces;140
8.4.4;4. Regular spaces, normal spaces;140
8.4.5;5. Accumulation points. Isolated points;142
8.4.6;6. The derived set;142
8.4.7;7. Sets dense in themselves;143
8.4.8;Exercises;144
8.5;CHAPTER XII. CONTINUOUS MAPPINGS;146
8.5.1;1. Continuity;146
8.5.2;2. Homeomorphisms;148
8.5.3;3. Case of metric spaces;150
8.5.4;4. Distance of a point from a set. Normality of metric spaces;155
8.5.5;5. Extension of continuous functions. Tietze theorem;157
8.5.6;6. Completely regular spaces;163
8.5.7;Exercises;164
8.6;CHAPTER XIII. CARTESIAN PRODUCTS;166
8.6.1;1. Cartesian product X x Y o f topological spaces;166
8.6.2;2. Projections and continuous mappings;167
8.6.3;3. Invariants of cartesian multiplication;168
8.6.4;4. Diagonal;169
8.6.5;5. Generalized cartesian products;170
8.6.6;6. XT considered as a topological space. The cube JT;171
8.6.7;7. Cartesian products of metric spaces;173
8.6.8;Exercises;174
8.7;CHAPTER XIV. SPACES WITH A COUNTABLE BASE;177
8.7.1;1. General properties;177
8.7.2;2. Separable spaces;178
8.7.3;3. Theorems on cardinality in spaces with countable bases;179
8.7.4;4. Imbedding in the Hilbert cube;180
8.7.5;5. Condensation points. The Cantor-Bendixson theorem;182
8.7.6;Exercises;184
8.8;CHAPTER XV. COMPLETE SPACES;186
8.8.1;1. Complete spaces;186
8.8.2;2. Cantor theorem;187
8.8.3;3. Baire theorem;187
8.8.4;4. Extension of a metric space to a complete space;189
8.8.5;Exercises;190
8.9;CHAPTER XVI. COMPACT SPACES;191
8.9.1;1. Definition;191
8.9.2;2. Fundamental properties of compact spaces;191
8.9.3;3. Cartesian products;193
8.9.4;4. Compactification of completely regular spaces;196
8.9.5;5. Compact metric spaces;198
8.9.6;6· The topology of uniform convergence of Yx;208
8.9.7;7. The compact-open topology of yx;209
8.9.8;8. The Cantor discontinuum;211
8.9.9;9. Continuous mappings of the Cantor discontinuum;214
8.9.10;Exercises;217
8.10;XVI. COMPACT SPACES;220
8.11;CHAPTER XVII. CONNECTED SPACES;224
8.11.1;1. Definition. Separated sets;224
8.11.2;2. Properties of connected spaces;226
8.11.3;3. Components;230
8.11.4;4. Cartesian products of connected spaces;232
8.11.5;5. Continua;233
8.11.6;6. Properties of continua;234
8.11.7;Exercises;238
8.12;CHAPTER XVIII. LOCALLY CONNECTED SPACES;241
8.12.1;1. Definitions and examples;241
8.12.2;2. Properties of locally connected spaces;241
8.12.3;3. Locally connected continua;244
8.12.4;4. Arcs. Arcwise connectedness;246
8.12.5;5. Continuous images of intervals;247
8.12.6;Exercises;253
8.13;CHAPTER XIX. THE CONCEPT OF DIMENSION;255
8.13.1;1. O-dimensional spaces;255
8.13.2;2. Properties of 0-dimensional metric separable spaces;255
8.13.3;3. n-dimensional spaces;256
8.13.4;4. Properties of n-dimensional metric separable spaces;258
8.13.5;Exercises;259
8.14;CHAPTER XX. SIMPLEXES AND THEIR PROPERTIES;260
8.14.1;1. Simplexes;260
8.14.2;2. Simplicial subdivision;262
8.14.3;3. Dimension of a simplex;266
8.14.4;4. The fixed point theorem;268
8.14.5;Exercises;271
8.15;CHAPTER XXI. CUTTINGS OF THE PLANE;274
8.15.1;1. Auxiliary properties of polygonal arcs;274
8.15.2;2. Cuttings;275
8.15.3;3. Complex functions which vanish nowhere. Existence of the logarithm;276
8.15.4;4. Auxiliary theorems;277
8.15.5;5. Corollaries to the auxiliary theorems;281
8.15.6;6. Theorems on the cuttings of the plane;283
8.15.7;7. Janiszewski theorems;285
8.15.8;8. Jordan theorem;286
8.15.9;Exercises;290
8.16;SUPPLEMENT: ELEMENTS OF ALGEBRAIC TOPOLOGY;292
8.16.1;Introduction;292
8.16.2;1. Complexes. Polyhedra. Simplicial approximation;293
8.16.3;2. Abelian groups;299
8.16.4;3. Categories and functors;304
8.16.5;4. Homology groups of simplicial complexes;307
8.16.6;5. Chain complexes;317
8.16.7;6. Homology groups of polyhedra;323
8.16.8;7. Homology groups with coefficients;330
8.16.9;8. Cohomology groups;334
8.16.10;Exercises;339
9;LIST OF IMPORTANT SYMBOLS;344
10;INDEX;346
11;OTHER TITLES IN THE SERIES IN PURE AND APPLIED MATHEMATICS;351