Metric Space Topology Examples, Exercises & Solutions by Wing-Sum Cheung [2025/2026 Study Guide Edition]

, 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 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
satisfyin
g
(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.