Topology : A First Course

by Munkres, James

• ISBN13: 9780139254956
• ISBN10: 0139254951
• Edition: 2nd
• Format: Hardcover
• Copyright: 2000-01-01
• Publisher: Pearson College Div

Table of Contents

Preface	xi	(4)
A Note to the Reader	xv
PART I	1	(226)
Chapter 1. Set Theory and Logic	3	(72)
1-1 Fundamental Concepts	4	(11)
1-2 Functions	15	(6)
1-3 Relations	21	(8)
1-4 The Integers and the Real Numbers	29	(7)
1-5 Arbitrary Cartesian Products	36	(3)
1-6 Finite Sets	39	(6)
1-7 Countable and Uncountable Sets	45	(8)
*1-8 The Principle of Recursive Definition	53	(4)
1-9 Infinite Sets and the Axiom of Choice	57	(6)
1-10 Well-Ordered Sets	63	(5)
*1-11 The Maximum Principle	68	(4)
*Supplementary Exercises: Well-Ordering	72	(3)
Chapter 2. Topological Spaces and Continuous Functions	75	(71)
2-1 Topological Spaces	75	(3)
2-2 Basis for a Topology	78	(6)
2-3 The Order Topology	84	(2)
2-4 The Product Topology on X x Y	86	(3)
2-5 The Subspace Topology	89	(3)
2-6 Closed Sets and Limit Points	92	(9)
2-7 Continuous Functions	101	(11)
2-8 The Product Topology	112	(5)
2-9 The Metric Topology	117	(9)
2-10 The Metric Topology (continued)	126	(8)
*2-11 The Quotient Topology	134	(10)
*Supplementary Exercises: Topological Groups	144	(2)
Chapter 3. Connectedness and Compactness	146	(43)
3-1 Connected Spaces	147	(5)
3-2 Connected Sets in the Real Line	152	(7)
*3-3 Components and Path Components	159	(2)
*3-4 Local Connectedness	161	(3)
3-5 Compact Spaces	164	(9)
3-6 Compact Sets in the Real Line	173	(5)
3-7 Limit Point Compactness	178	(4)
*3-8 Local Compactness	182	(5)
*Supplementary Exercises: Nets	187	(2)
Chapter 4. Countability and Separation Axioms	189	(38)
4-1 The Countability Axioms	190	(5)
4-2 The Separation Axioms	195	(12)
4-3 The Urysohn Lemma	207	(9)
4-4 The Urysohn Metrization Theorem	216	(6)
*4-5 Partitions of Unity	222	(3)
*Supplementary Exercises: Review of Part I	225	(2)
PART II	227	(172)
Chapter 5. The Tychonoff Theorem	229	(15)
5-1 The Tychonoff Theorem	229	(6)
5-2 Completely Regular Spaces	235	(3)
5-3 The Stone-Cech Compactification	238	(6)
Chapter 6. Metrization Theorems and Paracompactness	244	(18)
6-1 Local Finiteness	245	(2)
6-2 The Nagata-Smirnov Metrization Theorem (sufficiency)	247	(4)
6-3 The Nagata-Smirnov Theorem (necessity)	251	(3)
6-4 Paracompactness	254	(6)
6-5 The Smirnov Metrization Theorem	260	(2)
Chapter 7. Complete Metric Spaces and Function Spaces	262	(54)
7-1 Complete Metric Spaces	263	(8)
7-2 A Space-Filling Curve	271	(3)
7-3 Compactness in Metric Spaces	274	(6)
7-4 Pointwise and Compact Convergence	280	(5)
7-5 The Compact-Open Topology	285	(4)
7-6 Ascoli's Theorem	289	(4)
7-7 Baire Spaces	293	(4)
7-8 A Nowhere-Differentiable Function	297	(4)
7-9 An Introduction to Dimension Theory	301	(15)
Chapter 8. The Fundamental Group and Covering Spaces	316	(83)
8-1 Homotopy of Paths	318	(8)
8-2 The Fundamental Group	326	(5)
8-3 Covering Spaces	331	(5)
8-4 The Fundamental Group of the Circle	336	(7)
8-5 The Fundamental Group of the Punctured Plane	343	(5)
8-6 The Fundamental Group of S^n	348	(3)
8-7 Fundamental Groups of Surfaces	351	(6)
8-8 Essential and Inessential Maps	357	(4)
8-9 The Fundamental Theorem of Algebra	361	(3)
8-10 Vector Fields and Fixed Points	364	(5)
8-11 Homotopy Type	369	(5)
8-12 The Jordan Separation Theorem	374	(4)
8-13 The Jordan Curve Theorem	378	(9)
8-14 The Classification of Covering Spaces	387	(12)
Bibliography	399	(2)
Index	401