, 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 eff e 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 defi d metric d on X, we shall
simply call X a metric space.
1
,
,2 Metric Space Topology: Examples, Exercises and Solutions
Example 1.1.2.
(i) Let X := {a, b, c}, consider d1, d2 , and d3 : X ×X → R given by
d1 a b c d2 a b c d3 a b c
a 0 2 3 a 0 1 2 a 0 1 2
b 2 0 4 b 1 0 3 b 1 0 4
c 3 4 0 ; c 2 3 0 ; c 2 4 0 .
Then d1 and d2 are well-defi d metrics on X but d3 is not
(why?)
(ii) Let X be any nonempty set and d(x, y) := 1 − δxy for any x, y
∈ X, where δxy is the Kronecker delta function defi d by
0 if x = y
δxy := 1 if x = y .
Then d is a well-defined metric called the discrete metric and
(X, d) is called a discrete metric space. Notice that in a discrete
metric space, all distinct points have the same distance 1,
no matter how “far” or “close” they are from each other.
(iii) Let X := Rn and de(x, y) := x − y for any x, y ∈ Rn , where as
usual, stands for the Euclidean norm of Rn, that is,
n
z := z 2 ifor any z = (z 1 ,... , zn ) ∈ Rn . Hence
i=1
n
de(x, y) = (xi − yi)2
i=1
for any x = (x 1 ,... , xn), y = (y 1 ,... , yn) ∈ Rn. Then de is a
well-defi d metric called the Euclidean metric on Rn and (Rn ,
de) is called a Euclidean metric space. In general, when
we refer to the “Euclidean space Rn”, we mean the
Euclidean metric space (Rn, de ). In the sequel, unless it is
specifically mentioned, all Rn’s will be equipped with the
Euclidean metric de and so sometimes we will be sloppy and
write d for de and Rn for (Rn, de ) in case no confusion may
arise.
, Metric Spaces 3
(iv) Let X := C and dC(x, y) := |x − y| for any x, y ∈ C, where, as
usual, |z| := the modulus of z ∈ C. Then dC is a well-defined
metric, and we will see that (C, dC) is “equivalent” to the
Euclidean space (R2, de). In fact, it is easy to see that under
the
usual identification C ∼ = R2 , any x, y ∈ C can be considered as
points in R2, and under this identifi ation, dC(x, y) = de(x, y).
(v) Let X := R2 and d̃(x, y) := 100(x1 − y1 )2 + (x2 − y2 )2 for any x =
(x1 , x2 ) and y = (y1 , y2 ) ∈ R2 . Then d̃ is a well-defined metric on
X. Note that as real-valued functions of X × X into R, d̃ = de
.
(vi) Let X := S 1 ⊂ R2 and d(x, y) := the arc length of the smaller arc
in S1 joining x and y ∈ X. Then d is a well-defined metric on X.
Note that d = de as real-valued functions of X × X into R. This
is a typical example of “intrinsic metric” that can be defined
on a
metric space. It is called intrinsic as the distance between
two points can be measured without referring to the
ambient space R2. In fact, imagine you are an ant walking
on a rubber band. Then for you, the “distance” between
two points A, B on the rubber band is d(A, B) instead of
de(A, B), because the latter is the Euclidean distance
between the two points A, B which is attained by the
straight line segment joining the two points, but for you,
the ant, unless you can fly, you cannot travel from A to B along
the straight line segment joining the two points.
(vii) Let X := S 2 ⊂ R3 and x, y ∈ X. Observe that if x = ±y, then
there is a unique “great circle” on X passing through x,
y. Here, a great circle means a planar circle on S2 with
radius 1. Defi d(x, y) := the arc length of the smaller
arc on the unique great circle in S2 joining x and y. If
x = y, we
defi d(x, y) := 0. Finally, if x = −y, then they are antipodal
points on S2 and so they are joined by infi tely many great
circles. In this case we defi d(x, y) := π = half of the arc
length of any such great circles. Then, similar to Example (vi),
d is a well-defined intrinsic metric on X.
,4 Metric Space Topology: Examples, Exercises and Solutions
(viii) Let X := Rn and dS : X × X → R be defi d by
n
dS(x, y) := |xi − yi|
i=1
for any x = (x 1 ,... , xn), y = (y 1 ,... , yn) ∈ Rn . Then dS is a well-
defi d metric on Rn. Note that dS = de.
(ix) Let X := Rn and d : X × X → R be defi d by
∞
d∞(x, y) := max{|xi − y i| : i = 1, . . . , n}
for any x = (x 1 ,... , xn), y = (y 1 ,... , yn) ∈ Rn. Then d is a
∞
well-defi d metric on Rn. Note that de = d∞ = dS.
(x) Let X := Rn. For any k ∈ N, let dk : X × X → R be defi d by
1/k
n
k
|xi − yi|
dk(x, y) :=
i=1
for any x = (x 1 ,... , xn), y = (y 1 ,... , yn ) ∈ Rn. Then dk is a
well-defi d metric on Rn. Note that dk = de, dS , nor d ∞.
(xi) Let X := C[a, b ]. Clearly, X is a real vector space under point-
wise operations. For any f, g ∈ X, defi
b
d1(f, g) := |f − g|(x)dx ,
a
where, as usual,
|f − g|(x) := |f (x) − g(x)| for any x ∈ [a, b ] .
Then d1 is a well-defi d metric and so (X, d1) is a metric
space. [See Exercise 1.1, Part B, Problem #4.]
, Metric Spaces 5
(xii) Similar to Example (xi) above, let X := C[a, b ]. For any f, g
∈ X, defi
b |f − g| 1
2 2
d2(f, g) := (x)dx .
a
Then d2 is a well-defi d metric and (X, d2 ) is a metric space.
[See Exercise 1.1, Part B, Problem #4.]
(xiii) Again, let X := C[a, b ]. For any f, g ∈ X, defi
d∞(f, g) := sup{|f (x) − g(x)| : x ∈ [a, b ]} .
Then (X, d∞) is a metric space. Clearly, d∞ is diff ent from dp,
p = 1, 2, and as we will see in the sequel (see, for
example, Exercise 1.2, Part B, Problem #19 for details),
(X, d∞), (X, d1), and (X, d2) are very diff rent metric spaces.
(xiv) Let ℓ∞ be the space of all bounded sequences of real
numbers. So an element in ℓ∞ is of the form x = {xn }n∈N
with sup{|xn | : n ∈ N} < ∞. For any elements x = {xn}n∈N
and y = {yn }n∈N ∈ X, defi
d(x, y) := sup{|xn − yn| : n ∈ N} .
Then it is elementary to verify that (ℓ∞, d) is a metric space.
Remark. From Example 1.1.2, we see that on the same
nonempty set, different metrics could be defi d and they could in
turn make the same underlying set into metric spaces with very
different properties.
Definition 1.1.3. Let (X, d) be a metric space, x ∈ X, and A ⊂ X
be a nonempty subset. The distance from x to A is defined as
d(x, A) := inf d(x, a) : a ∈ A
, 6 Metric Space Topology: Examples, Exercises and Solutions
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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 eff e 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 defi d metric d on X, we shall
simply call X a metric space.
1
,
,2 Metric Space Topology: Examples, Exercises and Solutions
Example 1.1.2.
(i) Let X := {a, b, c}, consider d1, d2 , and d3 : X ×X → R given by
d1 a b c d2 a b c d3 a b c
a 0 2 3 a 0 1 2 a 0 1 2
b 2 0 4 b 1 0 3 b 1 0 4
c 3 4 0 ; c 2 3 0 ; c 2 4 0 .
Then d1 and d2 are well-defi d metrics on X but d3 is not
(why?)
(ii) Let X be any nonempty set and d(x, y) := 1 − δxy for any x, y
∈ X, where δxy is the Kronecker delta function defi d by
0 if x = y
δxy := 1 if x = y .
Then d is a well-defined metric called the discrete metric and
(X, d) is called a discrete metric space. Notice that in a discrete
metric space, all distinct points have the same distance 1,
no matter how “far” or “close” they are from each other.
(iii) Let X := Rn and de(x, y) := x − y for any x, y ∈ Rn , where as
usual, stands for the Euclidean norm of Rn, that is,
n
z := z 2 ifor any z = (z 1 ,... , zn ) ∈ Rn . Hence
i=1
n
de(x, y) = (xi − yi)2
i=1
for any x = (x 1 ,... , xn), y = (y 1 ,... , yn) ∈ Rn. Then de is a
well-defi d metric called the Euclidean metric on Rn and (Rn ,
de) is called a Euclidean metric space. In general, when
we refer to the “Euclidean space Rn”, we mean the
Euclidean metric space (Rn, de ). In the sequel, unless it is
specifically mentioned, all Rn’s will be equipped with the
Euclidean metric de and so sometimes we will be sloppy and
write d for de and Rn for (Rn, de ) in case no confusion may
arise.
, Metric Spaces 3
(iv) Let X := C and dC(x, y) := |x − y| for any x, y ∈ C, where, as
usual, |z| := the modulus of z ∈ C. Then dC is a well-defined
metric, and we will see that (C, dC) is “equivalent” to the
Euclidean space (R2, de). In fact, it is easy to see that under
the
usual identification C ∼ = R2 , any x, y ∈ C can be considered as
points in R2, and under this identifi ation, dC(x, y) = de(x, y).
(v) Let X := R2 and d̃(x, y) := 100(x1 − y1 )2 + (x2 − y2 )2 for any x =
(x1 , x2 ) and y = (y1 , y2 ) ∈ R2 . Then d̃ is a well-defined metric on
X. Note that as real-valued functions of X × X into R, d̃ = de
.
(vi) Let X := S 1 ⊂ R2 and d(x, y) := the arc length of the smaller arc
in S1 joining x and y ∈ X. Then d is a well-defined metric on X.
Note that d = de as real-valued functions of X × X into R. This
is a typical example of “intrinsic metric” that can be defined
on a
metric space. It is called intrinsic as the distance between
two points can be measured without referring to the
ambient space R2. In fact, imagine you are an ant walking
on a rubber band. Then for you, the “distance” between
two points A, B on the rubber band is d(A, B) instead of
de(A, B), because the latter is the Euclidean distance
between the two points A, B which is attained by the
straight line segment joining the two points, but for you,
the ant, unless you can fly, you cannot travel from A to B along
the straight line segment joining the two points.
(vii) Let X := S 2 ⊂ R3 and x, y ∈ X. Observe that if x = ±y, then
there is a unique “great circle” on X passing through x,
y. Here, a great circle means a planar circle on S2 with
radius 1. Defi d(x, y) := the arc length of the smaller
arc on the unique great circle in S2 joining x and y. If
x = y, we
defi d(x, y) := 0. Finally, if x = −y, then they are antipodal
points on S2 and so they are joined by infi tely many great
circles. In this case we defi d(x, y) := π = half of the arc
length of any such great circles. Then, similar to Example (vi),
d is a well-defined intrinsic metric on X.
,4 Metric Space Topology: Examples, Exercises and Solutions
(viii) Let X := Rn and dS : X × X → R be defi d by
n
dS(x, y) := |xi − yi|
i=1
for any x = (x 1 ,... , xn), y = (y 1 ,... , yn) ∈ Rn . Then dS is a well-
defi d metric on Rn. Note that dS = de.
(ix) Let X := Rn and d : X × X → R be defi d by
∞
d∞(x, y) := max{|xi − y i| : i = 1, . . . , n}
for any x = (x 1 ,... , xn), y = (y 1 ,... , yn) ∈ Rn. Then d is a
∞
well-defi d metric on Rn. Note that de = d∞ = dS.
(x) Let X := Rn. For any k ∈ N, let dk : X × X → R be defi d by
1/k
n
k
|xi − yi|
dk(x, y) :=
i=1
for any x = (x 1 ,... , xn), y = (y 1 ,... , yn ) ∈ Rn. Then dk is a
well-defi d metric on Rn. Note that dk = de, dS , nor d ∞.
(xi) Let X := C[a, b ]. Clearly, X is a real vector space under point-
wise operations. For any f, g ∈ X, defi
b
d1(f, g) := |f − g|(x)dx ,
a
where, as usual,
|f − g|(x) := |f (x) − g(x)| for any x ∈ [a, b ] .
Then d1 is a well-defi d metric and so (X, d1) is a metric
space. [See Exercise 1.1, Part B, Problem #4.]
, Metric Spaces 5
(xii) Similar to Example (xi) above, let X := C[a, b ]. For any f, g
∈ X, defi
b |f − g| 1
2 2
d2(f, g) := (x)dx .
a
Then d2 is a well-defi d metric and (X, d2 ) is a metric space.
[See Exercise 1.1, Part B, Problem #4.]
(xiii) Again, let X := C[a, b ]. For any f, g ∈ X, defi
d∞(f, g) := sup{|f (x) − g(x)| : x ∈ [a, b ]} .
Then (X, d∞) is a metric space. Clearly, d∞ is diff ent from dp,
p = 1, 2, and as we will see in the sequel (see, for
example, Exercise 1.2, Part B, Problem #19 for details),
(X, d∞), (X, d1), and (X, d2) are very diff rent metric spaces.
(xiv) Let ℓ∞ be the space of all bounded sequences of real
numbers. So an element in ℓ∞ is of the form x = {xn }n∈N
with sup{|xn | : n ∈ N} < ∞. For any elements x = {xn}n∈N
and y = {yn }n∈N ∈ X, defi
d(x, y) := sup{|xn − yn| : n ∈ N} .
Then it is elementary to verify that (ℓ∞, d) is a metric space.
Remark. From Example 1.1.2, we see that on the same
nonempty set, different metrics could be defi d and they could in
turn make the same underlying set into metric spaces with very
different properties.
Definition 1.1.3. Let (X, d) be a metric space, x ∈ X, and A ⊂ X
be a nonempty subset. The distance from x to A is defined as
d(x, A) := inf d(x, a) : a ∈ A
, 6 Metric Space Topology: Examples, Exercises and Solutions
THOSE WERE PREVIEW PAGES
TO DOWNLOAD THE FULL PDF
CLICK ON THE L.I.N.K
ON THE NEXT PAGE
