, Contents
Preface vii
A Note on the Convention xi
About the Author xiii
1. Metric Spaces 1
1.1 Definitions and Examples .............................................................. 1
Exercise 1.1: Part A ..................................................... 10
Exercise 1.1: Part B ..................................................... 14
1.2 Topology of Metric Spaces.......................................................... 36
Exercise 1.2: Part A ..................................................... 50
Exercise 1.2: Part B ..................................................... 64
1.3 Compactness ................................................................................ 85
Exercise 1.3: Part A ..................................................... 90
Exercise 1.3: Part B ..................................................... 94
1.4 Compactness in the Euclidean Space Rn ...................................................... 108
Exercise 1.4: Part A ................................................... 115
Exercise 1.4: Part B ................................................... 118
2. Limits and Continuity 129
2.1 Convergence in a Metric Space..................................................129
Exercise 2.1: Part A ................................................... 134
Exercise 2.1: Part B ................................................... 138
2.2 Complete Metric Spaces.............................................................145
Exercise 2.2: Part A ................................................... 150
Exercise 2.2: Part B ................................................... 155
2.3 Continuity and Homeomorphism ...............................................172
Exercise 2.3: Part A ................................................... 193
Exercise 2.3: Part B ................................................... 204
3. Connectedness 233
3.1 Connectedness.............................................................................233
Exercise 3.1: Part A ................................................... 245
Exercise 3.1: Part B ................................................... 249
Xv
,Xvi Metric Space Topology: Examples, Exercises and Solutions
3.2 Path-connectedness .................................................................... 266
Exercise 3.2: Part A.................................................... 278
Exercise 3.2: Part B.................................................... 281
4. Uniform Continuity 295
4.1 Uniform Continuity .................................................................... 296
Exercise 4.1: Part A.................................................... 301
Exercise 4.1: Part B.................................................... 309
4.2 Contraction and Beach’s Fixed Point Theorem...................... 322
Exercise 4.2: Part A.................................................... 330
Exercise 4.2: Part B.................................................... 332
5. Uniform Convergence 349
5.1 Sequence of Functions ................................................................ 349
Exercise 5.1: Part A.................................................... 368
Exercise 5.1: Part B.................................................... 377
5.2 Series of Functions ..................................................................... 389
Exercise 5.2: Part A.................................................... 395
Exercise 5.2: Part B.................................................... 401
Bibliography 421
Index 423
, Chapter 1
Metric Spaces
In this chapter, the basic concept of metric spaces will be introduced.
Naively, they are simply nonempty sets equipped with a structure
called metric. For the less matured students, at the beginning, this
structure may appear to be a bit abstract and difficult to master.
But in practice, this seemingly new concept is nothing more than
a tiny little abstractization of the familiar space Rn and so all one
needs to do is that whenever one needs to work on a problem in an
abstract metric space, one first looks at the problem on Rn, then one
would be able to see the clue of how to proceed in the general case.
In fact, in general, the most effective way to master a new concept
in any branch of mathematics is to keep in mind a couple of typical
concrete examples and think of these examples all the time. It is just
that easy.
1.1 Definitions and Examples
Definition 1.1.1. Let X be a nonempty set. A metric ton X is a
real-valued function
D: X × X → R
Satisfying
(M1) d(x, y) ≥ 0 and d(x, y) = 0 if and only if x = y,
(M2) (Symmetry) d(x, y) = d(y, x),
(M3) (Triangle inequality) d (x, y) ≤ d(x, z) + d (z, y)
For all x, y, z ∈ X. Given x, y ∈ X, d(x, y) is also known as the
distance between x and y with respect to do. The pair (X, d) is
called a metric space and elements in X are referred to as points in
X. For the sake of convenience, in case there is a clearly defined
metric d on X, we shall simply call X a metric space.
1
Preface vii
A Note on the Convention xi
About the Author xiii
1. Metric Spaces 1
1.1 Definitions and Examples .............................................................. 1
Exercise 1.1: Part A ..................................................... 10
Exercise 1.1: Part B ..................................................... 14
1.2 Topology of Metric Spaces.......................................................... 36
Exercise 1.2: Part A ..................................................... 50
Exercise 1.2: Part B ..................................................... 64
1.3 Compactness ................................................................................ 85
Exercise 1.3: Part A ..................................................... 90
Exercise 1.3: Part B ..................................................... 94
1.4 Compactness in the Euclidean Space Rn ...................................................... 108
Exercise 1.4: Part A ................................................... 115
Exercise 1.4: Part B ................................................... 118
2. Limits and Continuity 129
2.1 Convergence in a Metric Space..................................................129
Exercise 2.1: Part A ................................................... 134
Exercise 2.1: Part B ................................................... 138
2.2 Complete Metric Spaces.............................................................145
Exercise 2.2: Part A ................................................... 150
Exercise 2.2: Part B ................................................... 155
2.3 Continuity and Homeomorphism ...............................................172
Exercise 2.3: Part A ................................................... 193
Exercise 2.3: Part B ................................................... 204
3. Connectedness 233
3.1 Connectedness.............................................................................233
Exercise 3.1: Part A ................................................... 245
Exercise 3.1: Part B ................................................... 249
Xv
,Xvi Metric Space Topology: Examples, Exercises and Solutions
3.2 Path-connectedness .................................................................... 266
Exercise 3.2: Part A.................................................... 278
Exercise 3.2: Part B.................................................... 281
4. Uniform Continuity 295
4.1 Uniform Continuity .................................................................... 296
Exercise 4.1: Part A.................................................... 301
Exercise 4.1: Part B.................................................... 309
4.2 Contraction and Beach’s Fixed Point Theorem...................... 322
Exercise 4.2: Part A.................................................... 330
Exercise 4.2: Part B.................................................... 332
5. Uniform Convergence 349
5.1 Sequence of Functions ................................................................ 349
Exercise 5.1: Part A.................................................... 368
Exercise 5.1: Part B.................................................... 377
5.2 Series of Functions ..................................................................... 389
Exercise 5.2: Part A.................................................... 395
Exercise 5.2: Part B.................................................... 401
Bibliography 421
Index 423
, Chapter 1
Metric Spaces
In this chapter, the basic concept of metric spaces will be introduced.
Naively, they are simply nonempty sets equipped with a structure
called metric. For the less matured students, at the beginning, this
structure may appear to be a bit abstract and difficult to master.
But in practice, this seemingly new concept is nothing more than
a tiny little abstractization of the familiar space Rn and so all one
needs to do is that whenever one needs to work on a problem in an
abstract metric space, one first looks at the problem on Rn, then one
would be able to see the clue of how to proceed in the general case.
In fact, in general, the most effective way to master a new concept
in any branch of mathematics is to keep in mind a couple of typical
concrete examples and think of these examples all the time. It is just
that easy.
1.1 Definitions and Examples
Definition 1.1.1. Let X be a nonempty set. A metric ton X is a
real-valued function
D: X × X → R
Satisfying
(M1) d(x, y) ≥ 0 and d(x, y) = 0 if and only if x = y,
(M2) (Symmetry) d(x, y) = d(y, x),
(M3) (Triangle inequality) d (x, y) ≤ d(x, z) + d (z, y)
For all x, y, z ∈ X. Given x, y ∈ X, d(x, y) is also known as the
distance between x and y with respect to do. The pair (X, d) is
called a metric space and elements in X are referred to as points in
X. For the sake of convenience, in case there is a clearly defined
metric d on X, we shall simply call X a metric space.
1
