The material in this book is organized in such a way that the reader gets to significant applications quickly, and the emphasis is on the geometric understanding and use of new concepts. The theme of the book is that topology is really the language of modern mathematics.

lgli/Borges C.R. Elementary topology and applications (WS, 2001)(ISBN 9810242409)(O)(215s)_MDgt_.djvu

lgrsnf/Borges C.R. Elementary topology and applications (WS, 2001)(ISBN 9810242409)(O)(215s)_MDgt_.djvu

zlib/Mathematics/Carlos R. Borges/Elementary topology and applications_538436.djvu

Borges, Carlos R

World Scientific Publishing Co Pte Ltd

Singapore, River Edge, NJ, Singapore, 2000

Singapore, cop. 2000

January 15, 2001

1

lg110224

{"isbns":["9789810242404","9789812386465","9810242409","9812386467"],"last_page":215,"publisher":"WORLD SCIENTIFIC"}

Includes index

Bookmarks: p1 (p1): CHAPTER 0. SETS AND NUMBERS
p1-1 (p1): 0.1 Rudiments of Logic
p1-2 (p5): 0.2 Fundamentals of Set Description
p1-3 (p5): 0.3 Set Inclusion and Equality
p1-4 (p6): 0.4 An Axiom System for Set Theory
p1-5 (p6): 0.5 Unions and Intersections
p1-6 (p7): 0.6 Set Difference
p1-7 (p7): 0.7 Integers and Induction
p1-8 (p11): 0.8 Simple Cartesian Products
p1-9 (p11): 0.9 Relations
p1-10 (p12): 0.10 Functions
p1-11 (p14): 0.11 Sequences
p1-12 (p15): 0.12 Indexing Sets
p1-13 (p16): 0.13 Important Formulas
p1-14 (p16): 0.14 Inverse Functions
p1-15 (p17): 0.15 More Important Formulas
p1-16 (p18): 0.16 Partitions
p1-17 (p18): 0.17 Equivalence Relations, Partitions and Functions
p1-18 (p19): 0.18 General Cartesian Products
p1-19 (p20): 0.19 The Sixth Axiom (Axiom of Choice)
p1-20 (p21): 0.20 Well-Orders and Zorn
p1-21 (p22): 0.21 Yet More Important Formulas
p1-22 (p22): 0.22 Cardinality
p2 (p31): CHAPTER 1. METRIC AND TOPOLOGICAL SPACES
p2-1 (p31): 1.1 Metrics and Topologies
p2-2 (p33): 1.2 Time out for Notation
p2-3 (p35): 1.3 Metrics Generate Topologies
p2-4 (p36): 1.4 Continuous Functions
p2-5 (p39): 1.5 Subspaces
p2-6 (p39): 1.6 Comparable Topologies
p3 (p47): CHAPTER 2. FROM OLD TO NEW SPACES
p3-1 (p47): 2.1 Product Spaces
p3-2 (p51): 2.2 Product Metrics and Topologies
p3-3 (p53): 2.3 Quotient Spaces
p3-4 (p55): 2.4 Applications (Mobius Band, Klein Bottle, Torus, Projective Plane, etc.)
p4 (p67): CHAPTER 3. VERY SPECIAL SPACES
p4-1 (p67): 3.1 Compact Spaces
p4-2 (p73): 3.2 Compactif'ication (One-Point Only)
p4-3 (p75): 3.3 Complete Metric Spaces (Baire-Category, Banach Contraction Theorem and Applications of Roots of y = h(x) to Systems of Linear Equations
p4-4 (p80): 3.4 Connected and Arcwise Connected Spaces
p5 (p89): CHAPTER 4. FUNCTION SPACES
p5-1 (p89): 4.1 Function Space Topologies
p5-2 (p92): 4.2 Completness and Compactness (Ascoli-Arzela Theorem, Picard's Theorem, Peano's Theorem)
p5-3 (p100): 4.3 Approximation (Bernstein's polynomials, Stone-Weierstrass Approximation)
p5-4 (p103): 4.4 Function-Space Functions
p6 (p114): CHAPTER 5. TOPOLOGICAL GROUPS
p6-1 (p114): 5.1 Elementary Structures
p6-2 (p121): 5.2 Topological Isomorphism Theorems
p6-3 (p123): 5.3 Quotient Group Recognition
p6-4 (p124): 5.4 Morphism Groups (Topological and Transformation Groups)
p7 (p131): CHAPTER 6. SPECIAL GROUPS
p7-1 (p131): 6.1 Preliminaries
p7-2 (p134): 6.2 Groups of Matrices
p7-3 (p135): 6.3 Groups of Isometries
p7-4 (p140): 6.4 Relativity and Lorentz Transformations
p8 (p147): CHAPTER 7. NORMALITY AND PARACOMPACTNESS
p8-1 (p147): 7.1 Normal Spaces (Urysohn's Lemma)
p8-2 (p151): 7.2 Paracompact Spaces (Partitions of Unity with and Application to Embedding of Manifolds in Euclidean Spaces)
p9 (p167): CHAPTER 8. THE FUNDAMENTAL GROUP
p9-1 (p167): 8.1 Description of II,(X,b)
p9-2 (p173): 8.2 Elementary Facts about II,(X,b)
p9-3 (p175): 8.3 Simplicial Complexes
p9-4 (p179): 8.4 Barycentric Subdivision
p9-5 (p181): 8.5 The Simplicial Approximation
p9-6 (p183): 8.6 The Fundamental Group of Polytopes
p9-7 (p187): 8.7 Graphs and Trees
p10 (p193): APPENDIX A. SOME INEQUALITIES
p11 (p195): APPENDIX B. BINOMIAL EQUALITIES
p12 (p197): LIST OF SYMBOLS
p13 (p199): INDEX

This text argues that topology is the language of modern mathematics. Material in the book is organized in such a way that the reader reaches significant applications of this premise quickly, emphasizing the geometric understanding and use of new concepts.

The main purpose of this section is to collect those precious gems of Set Theory - Relations, Functions and Inverse Functions - which will frequently be used throughout the text.

2009-08-06