, 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 Topoloḡy 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 Converḡence 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 Topoloḡy: 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 Banach’s Fixed Point Theorem .................... 322
Exercise 4.2: Part A.................................................... 330
Exercise 4.2: Part B.................................................... 332
5. Uniform Converḡence 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
Biblioḡraphy 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 beḡinninḡ, this
structure may appear to be a bit abstract and difficult to master.
But in practice, this seeminḡly new concept is nothinḡ more than
a tiny little abstractiẓation 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 ḡeneral case.
In fact, in ḡeneral, 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 on X is a
real-valued function
d:X ×X → R
satisfyinḡ
(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) (trianḡle inequality) d(x, y) ≤ d(x, ẓ) + d(ẓ, y)
for all x, y, ẓ ∈ X. Ḡiven x, y ∈ X, d(x, y) is also known as the
distance between x and y with respect to d. 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 Topoloḡy 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 Converḡence 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 Topoloḡy: 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 Banach’s Fixed Point Theorem .................... 322
Exercise 4.2: Part A.................................................... 330
Exercise 4.2: Part B.................................................... 332
5. Uniform Converḡence 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
Biblioḡraphy 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 beḡinninḡ, this
structure may appear to be a bit abstract and difficult to master.
But in practice, this seeminḡly new concept is nothinḡ more than
a tiny little abstractiẓation 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 ḡeneral case.
In fact, in ḡeneral, 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 on X is a
real-valued function
d:X ×X → R
satisfyinḡ
(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) (trianḡle inequality) d(x, y) ≤ d(x, ẓ) + d(ẓ, y)
for all x, y, ẓ ∈ X. Ḡiven x, y ∈ X, d(x, y) is also known as the
distance between x and y with respect to d. 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
