Intro
Foreword
Preface
Contents
1 Real Analysis
1.1 The Real Number System
1.2 Sequences of Real Numbers
1.2.1 Convergence of a Sequence
1.2.2 Algebra of Limits
1.2.3 Bounded Monotone Sequences
1.2.4 Cauchy Sequences
1.3 Series Convergence
1.4 Decimal and General Expansions
1.5 Continuity
1.6 Uniform Convergence
1.6.1 Necessary and Sufficient Conditions
1.6.2 Notes and Remarks
1.7 Hints and Solutions to Selected Exercises
References
2 Metric Spaces
2.1 Introduction
2.1.1 The Euclidean Spaces
2.1.2 Balls and Bounded Sets

2.2 Convergence in Metric Spaces
2.3 Normed Linear Spaces
2.4 Sequence Spaces
2.5 Hints and Solutions to Selected Exercises
References
3 Topology
3.1 Open Sets and Closed Sets
3.2 Limit Points and Isolated Points
3.3 Closures and Boundaries
3.4 Subspace Topology
3.5 Limits and Continuity
3.5.1 The Case of Euclidean Spaces
3.5.2 Continuity and Uniform Convergence
3.6 Topology of Normed Linear Spaces
3.7 Hints and Solutions to Selected Exercises
References
4 Completeness
4.1 Introduction
4.2 Banach Contraction Principle

4.3 Characterizations of Completeness
4.3.1 Cantor Intersection Property
4.3.2 Totally Bounded Sets
4.4 Completion of a Metric Space
4.5 Banach Spaces
4.6 Hints and Solutions to Selected Exercises
References
5 Compactness
5.1 Introduction
5.1.1 Compact Sets and Closed Sets
5.1.2 Compact Subsets of Euclidean Spaces
5.2 Characterizations of Compact Sets
5.2.1 Finite Intersection Property
5.2.2 Sequentially Compact Sets
5.3 Continuity and Compactness
5.3.1 Uniform Continuity
5.3.2 Notes and Remarks
5.4 Lipschitz Continuity

5.5 Hints and Solutions to Selected Exercises
References
6 Connectedness
6.1 Path Connectedness
6.2 Connected Sets
6.3 Components
6.4 Miscellaneous
6.4.1 Locally Connected and Locally Path Connected Spaces
6.4.2 Path Connectedness in Locally Path Connected Spaces
6.4.3 Quasi-components
6.4.4 Totally Disconnected Sets
6.5 Hints and Solutions to Selected Exercises
References
7 Cardinality
7.1 Countable and Uncountable Sets
7.2 Some Applications to Topology
7.3 The Set of Discontinuities
7.3.1 The Case of Monotone Functions
7.3.2 The General Case

7.4 Cardinality
7.4.1 Cardinal Numbers
7.4.2 Notes and Remarks
7.5 Hints and Solutions to Selected Exercises
References
8 Denseness
8.1 Separability
8.2 Perfect Sets
8.3 Baire Category Theorem
8.4 Equicontinuity
8.5 Hints and Solutions to Selected Exercises
References
9 Homeomorphisms
9.1 Equivalent Metrics
9.2 Homeomorphisms
9.3 Extension Theorems for Continuous Functions
9.4 Finite-Dimensional Normed Linear Spaces
9.5 Hints and Solutions to Selected Exercises
References
10 The Cantor Set
10.1 Introduction