An Introduction to Metric Spaces

Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface
A Note to the Reader
Authors
1: Set Theory
	1.1 Sets
		1.1.1 The empty set
		1.1.2 Operations on sets
		1.1.3 Uniqueness of the empty set
		1.1.4 Power sets
		1.1.5 Cartesian products
	1.2 Relations
		1.2.1 Types of relations
		1.2.2 Equivalence relations
		1.2.3 Partition of sets
		1.2.4 Partial order relations
	1.3 Functions
		1.3.1 Composition of functions
		1.3.2 Inverse of a function
		1.3.3 Images of sets under functions
		1.3.4 Inverse images of sets under functions
	1.4 Countability of Sets
		1.4.1 Finite sets
		1.4.2 Countable sets
	Problem Set
	Biographical Notes
2: Metric Spaces
	2.1 Review of Real Number System and Absolute Value
	2.2 Young, Hölder, and Minkowski Inequalities
	2.3 Notion of Metric Space
	2.4 Open Sets
		2.4.1 Subspace topology
		2.4.2 Product topology
	2.5 Closed Sets
	2.6 Interior, Exterior, and Boundary Points
	2.7 Limit and Cluster Points
	2.8 Bounded Sets
	2.9 Distance Between Sets
	2.10 Equivalent Metrics
	Problem Set
	Biographical Notes
3: Complete Metric Spaces
	3.1 Sequences
		3.1.1 Subsequences
	3.2 Convergence of Sequence
	3.3 Complete Metric Spaces
	3.4 Completion of Metric Spaces
		3.4.1 Construction of the set Z
		3.4.2 Embedding X in Z
		3.4.3 Proving Z is complete
		3.4.4 Uniqueness of extension up to isometry
	3.5 Baire Category Theorem
		3.5.1 Category of sets
		3.5.2 Baire category theorem
		3.5.3 Applications of Baire category theorem
	Problem Set
	Biographical Notes
4: Compact Metric Spaces
	4.1 Open Cover and Compact Sets
	4.2 General Properties of Compact Sets
	4.3 Sufficient Conditions for Compactness
	4.4 Sequential Compactness
	4.5 Compactness: Characterizations
	Problem Set
	Biographical Notes
5: Connected Spaces
	5.1 Connectedness
		5.1.1 Connected subsets
	5.2 Components
	5.3 Totally Disconnected Spaces
	Problem Set
6: Continuity
	6.1 Continuity of Real Valued Functions
	6.2 Continuous Functions in Arbitrary Metric Spaces
		6.2.1 Equivalent definitions of continuity and other characterizations
		6.2.2 Results on continuity
	6.3 Uniform Continuity
	6.4 Continuous Functions on Compact Spaces
	6.5 Continuous Functions on Connected Spaces
		6.5.1 Path connectedness
	6.6 Equicontinuity and Arzela-Ascoli’s Theorem
	6.7 Open and Closed Maps
	6.8 Homeomorphism
	Problem Set
	Biographical Notes
7: Banach Fixed Point Theorem and Its Applications
	7.1 Banach Contraction Theorem
	7.2 Applications of Banach Contraction Principle
		7.2.1 Root finding problem
		7.2.2 Solution of systemof linear algebraic equations
		7.2.3 Picard existence theorem for differential equations
		7.2.4 Solutions of integral equations
		7.2.5 Solutions of initial value and boundary value problems
		7.2.6 Implicit function theorem
	Problem Set
	Biographical Notes
Appendix A
Bibliography
Index