A Guide to Topology (Dolciani Mathematical Expositions)

__A Guide to Topology__ is an introduction to basic topology. It covers point-set topology as well as Moore-Smith convergence and function spaces. It treats continuity, compactness, the separation axioms, connectedness, completeness, the relative topology, the quotient topology, the product topology, and all the other fundamental ideas of the subject. The book is filled with examples and illustrations. Graduate students studying for the qualifying exams will find this book to be a concise, focused and informative resource. Professional mathematicians who need a quick review of the subject, or need a place to look up a key fact, will find this book to be a useful research too. Cover 1 Copyright page 3 Title page 4 Preface 8 Contents 12 1 Fundamentals 14 1.1 What is Topology? 14 1.2 First Definitions 14 1.3 Mappings 21 1.4 The Separation Axioms 24 1.5 Compactness 30 1.6 Homeomorphisms 34 1.7 Connectedness 37 1.8 Path-Connectedness 40 1.9 Continua 41 1.10 Totally Disconnected Spaces 43 1.11 The Cantor Set 46 1.12 Metric Spaces 49 1.13 Metrizability 53 1.14 Baire’s Theorem 54 1.15 Lebesgue’s Lemma and Lebesgue Numbers 58 2 Advanced Properties of Topological Spaces 60 2.1 Basis and Subbasis 60 2.2 Product Spaces 61 2.3 Relative Topology 64 2.4 First Countable, Second Countable, and So Forth 64 2.5 Compactifications 67 2.6 Quotient Topologies 69 2.7 Uniformities 71 2.8 Morse Theory 73 2.9 Proper Mappings 78 2.10 Paracompactness 79 3 Moore-Smith Convergence and Nets 82 3.1 Introductory Remarks 82 3.2 Nets 82 4 Function Spaces 86 4.1 Preliminary Ideas 86 4.2 The Topology of Pointwise Convergence 87 4.3 The Compact-Open Topology 88 4.4 Uniform Convergence 89 4.5 Equicontinuity and the Ascoli-Arzela Theorem 91 4.6 The Weierstrass Approximation Theorem 94 Table of Notation 100 Glossary 104 Bibliography 112 Index 116 About the Author 120 A Guide to Topology is an introduction to basic topology. It covers point-set topology as well as Moore-Smith convergence and function spaces. It treats continuity, compactness, the separation axioms, connectedness, completeness, the relative topology, the quotient topology, the product topology, and all the other fundamental ideas of the subject. The book is filled with examples and illustrations. Graduate students studying for the qualifying exams will find this book to be a concise, focused and informative resource. Professional mathematicians who need a quick review of the subject, or need a place to look up a key fact, will find this book to be a useful research too. Steven Krantz is well-known for his skill in expository writing and this volume confirms it. He is the author of more than 50 books, and more than 150 scholarly papers. The MAA has awarded him both the Beckenbach Book Prize and the Chauvenet Prize

A Guide to Topology is an introduction to basic topology for graduate or advanced undergraduate students. It covers point-set topology, Moore-Smith convergence and function spaces. It treats continuity, compactness, the separation axioms, connectedness, completeness, the relative topology, the quotient topology, the product topology, and all the other fundamental ideas of the subject. The book is filled with examples and illustrations. Students studying for exams will find this book to be a concise, focused and informative resource. Professional mathematicians who need a quick review of the subject, or need a place to look up a key fact, will find this book to be a useful resource too.