Metric Space Topology: Examples, Exercises and Solutions (Cheung, 2024)

, 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


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,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 Banach’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 on 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 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.


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