Download Every Compact Metric Space is a Continuous Image of The Cantor Set

Every Compact Metric Space is a
Continuous Image of The Cantor Set
Rakesh Jana
132123025
Department of Mathematics
Indian Institute of Technology Guwahati
Abstract
Cantor set plays an important role in many branches of mathematics, and is not just an
artificial construct, especially designed to exhibit the possible pathologies that can arise
in the systematic development of real analysis.
Eighty year ago, Felix Hausdorff and Paul Alexandroff published independently
a theorem assertaing that every compact metric space is a continuous image of the
Cantor set. This theorem found its application in various branches of mathematics and
played also an importent role in the theory of curves and functional analysis. In this
article we discuss an importent application of Hausdorff-Alexandroff Theorem, called
Banach-Mazur Theorem.
List of Symbols
∩
∪
⊆
∈
ε
C
I
K
N
R
∑k
∏
kxk
πn
X uY
f ◦g
X∗
BX ∗
C([0, 1])
C(K)
intersection
union
subset
belongs to
epsilon
Cantor Set
[0,1]
scalar field
set of natural number
set of all real number
summation over k = 1 to ∞
product
norm of an element x in a normed space
projection map
X is homeomorphic to Y
composition of two function f and g
dual of X
closed unit call of X ∗ ; i.e. {x∗ ∈ X ∗ : kx∗ k ≤ 1}
set of all continuous function from [0, 1] to K
set of all continuous function from K to K
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1
Introduction
2
Cantor Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1
Construction of Cantor Set
6
2.2
Definition
7
2.3
Properties
7
3
Hausdorff - Alexandroff Theorem . . . . . . . . . . . . . . . . . . . . . . . 9
3.1
Idea of the proof
3.2
Hausdorff - Alexandroff Theorem
4
Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.1
Some important results and definitions
11
4.2
Banach-Mazur Theorem
12
5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5
9
10
Introduction
1. Introduction
1.1
Introduction
The basic concepts of topology(or analysis situs, as it was called at very beginning of its
history) were established during the development of set theory and process of rigorization
of analysis in the nineteenth century. Topology as separate branch of mathematics was
developed at the beginning of the twentieth century.
Significant contribution in set-theoretic topology was made namely by Felix Hausdroff. His book Grundzuge der Mengenlehre was published in 1914 and soon became a
classical work of set theory and topology.
Compactness is one of the central notions in topology. An early formulation of
compactness says that space is compact if every infinite set in the space has a limit point.
Further development of topology adopted more general definition of compactness in
terms of open coverings of the space. It is based on an abstraction of a certain property
of the set of real numbers described by Heine- Borel - Lebesgue theorem. A topological
space is said to be compact if and only if each open covering contains a finite sub
covering.
The ground work in the field of compact spaces was done by P. S. Alexandroff and P.
S. Urysohn. An important example of a compact metric space is the Cantor set. It was
introduced by Georg Cantor as an example of a perfect nowhere dense set. Special role
of the Cantor set in theory of compact spaces is given also by the fact that any compact
metric space is a continuous image of the Cantor set. This theorem appeared for first
time in 1927. It is sometime reffered to as the Hausdorff theorem but it is more accurate
to call it the Hausdorff - Alxendroff theorem since it is an example of a mathematical
theorem that was proved independently by two mathematician.
The theorem appeared in the second edition of Felix Hausdorff’s Mengenlehre and
in the article by P.S. Alexandroff published in Mathematischen Annalen int the same
year (1927).
Construction of Cantor Set
Definition
Properties
2. Cantor Set
Cantor set plays an important role in many branches of mathematics, and is not just an
artificial construct, especially designed to exhibit the possible pathologies that can arise
in the systematic development of real analysis.
2.1
Construction of Cantor Set
The cantor set has many definitions and many different construction. Although Georg
Cantor originally provided a purely abstract definition in 1883, the most accessible
is the Cantor "middle-thirds" or ternary set construction. Begin with the closed real
interval I = [0, 1] and
divide it into three equal open sub intervals. Remove the central
1 2
open interval 3 , 3 to obtain the closed set C1 ,which is union of 2 disjoint closed
intervals,each of lengths 13 .
1 [ 2
C1 = 0,
,1
3
3
Next, subdivide each of these two remaining intervals into three equal open sub intervals
and from each remove the central third and we obtain C2 which is the union of 22 closed
intervals, each length 312 .
1 [ 2 3 [ 6 7 [ 8
C2 = 0, 2
,
,
,1 .
3
32 32
32 32
32
We now repeat this "open middle one-third removal" on each of four intervals in C2 to
obtain C3 which is the union of 23 closed intervals, each length 313 . We continue this
removal operation countably many times to obtain the countable collection set {Ck }∞
k=1 .
2.2 Definition
2.2
7
Definition
The cantor set C is defined as
C=
∞
\
Ck
k=1
The collection {Ck }∞
k=1 possesses the following two condition:
∞
1. {Ck }k=1 is a descending sequence of closed sets;
2. For each k, Ck is the disjoint union of 2k closed intervals, each length
1
.
3k
Lemma 2.2.1 Every number in (0, 1) has a ternary expansion.
Definition 2.2.1 — Cantor set. Let C be the cantor set of the middle thirds:
(
C :=
2.3
)
ak
x ∈ R : x = ∑ k , whereak is either 0 or 2
k 3
Properties
Proposition 2.3.1 Every ternary expansion of number (0, 1) either unique or we have
exactly two representation of the form
x = .a1 a2 · · · aN−1 aN 222 · · · or x = .a1 a2 · · · aN−1 (aN + 1)000 · · · ,
for some N ∈ N.
Lemma 2.3.2 The unit interval [0, 1] is the continuous image of the cantor set.
k
Proof. Consider the map g : C → [0, 1] is given by g ∑k 3akk = ∑k 2ak+1
.
First we show that g is well defined . Let x, y ∈ C and suppose x = y. Then from
Lemma-2.2.1, x, y have ternary expansions
∞
x=
∞
ak
b
and
y
=
∑ 3k
∑ 3kk
k=1
k=1
where ak , bk = 0 or 2.
Case-1: If ak = bk for all k, then clearly g(x) = g(y).
Case-2: If ak = bk not for all k then by Proposition-2.3.1 there exist some N ∈ N such
that
x = .a1 a2 · · · aN−1 aN 222 · · · and y = .a1 a2 · · · aN−1 (aN + 1)000 · · · ,
That is ai = bi for i = 1, 2, · · · , (N − 1) and bN = aN + 1 , but aN and bN are either 0 or 2
, so bN = aN + 1 is not possible. Thus there does not exist any N ∈ N such that aN 6= bN .
Hence g is well defined.
Next, we show that g is continuous on C. Let ε > 0 be given. Then let x ∈ C and let
{xn }∞
n=0 be a sequence in C that converge to x. Now choose a natural number N so that
N
1/2 < ε. Since {xn }∞
n=0 converge to x, there exist a natural number M > N such that
|xn − x| < 1/3M for n > M. Then x and xn must lie in the same interval in Ck for all
Chapter 2. Cantor Set
8
k > M. Hence the ternary expansion choosing those with all 0s and 2s of x and xn must
agree for the first M terms. Thus
∞
|g(xn ) − g(x)| <
1
1
1
1
1
= M < N < ε,
= M+1
1
k
2
2
2
1− 2
k=M+1 2
∑
∀n > M.
Hence, {g(xn )}∞
n=0 converge to g(x). Therefore, g is continuous on C.
Lemma 2.3.3 The cantor set C is homemorphic to ∏N {0, 2}, the countable product of
the two point space 0,2 with discrete topology.
Proof. Consider
where Xn = {0, 2}
themap h : C → ∏N Xn ,
ak
defined as, h ∑k 3k = (a1 , a2 , · · · ).
Clearly h is one-one and onto. To show that h is continuous at x ∈ C, it is sufficient
to show that for each n , the coordinate function πn ◦ h is continuous. Now, πn ◦ h is
constant on the neighborhood of x in C. So πn ◦ h is continuous at x.
Again each Xn are Hausdorff. Thus ∏∞
1 Xn is Hausdorff.
Since h is continuous bijection from compact space to Hausdorff space. So, h is
homemorphic. Hence the proof.
Lemma 2.3.4 The cantor set is homeomorphic to the countable product of cantor set.
Proof.
CN u ({0, 2}N )N u {0, 2}NN u {0, 2}N u C
Idea of the proof
Hausdorff - Alexandroff Theorem
3. Hausdorff - Alexandroff Theorem
Theorem 3.0.5 Every compact metric space is a continuous image of the Cantor set.
3.1
Idea of the proof
The proof is just combine the following three facts
• Every compact metric space is homeomorphic to a subset of the Hilbert cube I N .
• The Hilbert cube is a continuous image of the Cantor set.
• Every closed subset of a Cantor set is the continuous image of the Cantor set.
We now prove the above three facts:
Lemma 3.1.1 If (x, d) is a compact metric space, then X is a homeomorphic to a subset
of I N
Proof. We may and do assume that the metric on X is bounded by 1. Since X is a
compact metric space, there exist a countable dense subset, say {xn : n ∈ N}. We define
F : X → I N by setting
F(x) := (d(x, x1 ), d(x, x2 ), · · · d(x, xn ), · · · ) .
The coordinate function πn ◦ F :→ I are continuous. Hence F is continuous. We claim
that F is one-one. Suppose that x, y ∈ X are such that F(x) = F(y). Since {xn } is a
sequence in a compact metric space X, so it has a convergent sub-sequence in X. Let
{xnk } be the convergent sub-sequence of {xn } and converge to x. Hence d (xnk , x) → 0
as k → ∞. Since F(x) = F(y), it follows that d (x, xn ) = d (y, xn ) ∀n. In particular
d (x, xnk ) = d (y, xnk ) → 0. Since the limit of the sequence in a metric space is unique ,
We conclude that x = y. This establish our claim. Since X is compact and I N is Hausdorff,
it follows that F : X → F(X) is a homeomorphism.
Lemma 3.1.2 The Hilbert cube I ∞ is the continuous image of the cantor set.
10
Chapter 3. Hausdorff - Alexandroff Theorem
Proof. In the view of Lemma-2.3.4, we may assume that any x ∈ C is of the form
(x1 , x2 , · · · ) , xi ∈ C. We consider a map G from CN to I N , define by, G(x) = (g(x1 ), · · · , g(xn ), · · · ),
where g is as in the proof of Lemma-2.3.2.
Clearly this map is one-one and onto. It is also continuous as its coordinate function
are continuous. Again G is a continuous bijection from the compact space to Hausdorff
space(as I N is Hausdorff). So G is a homeomorphism. Hence the proof.
Lemma 3.1.3 If K is a closed subset of the cantor set C, then K is the continuous image
of the cantor set.
Proof. Let the middle-two-thirds set C0 be the set of real number of the form ∑k b6kk where
bk is either 0 or 5. The obvious, as seen in Lemma-2.3.3, it is the homeomorphic to
0
∏∞
1 {0, 2}. Hence the cantor set C and the middle-two-third sets C are homeomorphic.
0
0
The set C has the property that if x, y ∈ C then their mid point (x + y)/2 does not lie
in C0 . Now assume that K 0 is a closed subset of C0 . If x0 ∈ C0 then there exist a unique
point kx ∈ K 0 such that d(x0 , kx ) = d(x0 , K 0 ). The function k : C0 → K 0 given by k(x) = kx
is a continuous, onto retraction.
3.2
Hausdorff - Alexandroff Theorem
Theorem 3.2.1 Every compact metric space is a continuous image of the Cantor set.
Proof. Let us assume that the given compact metric space X. Then there exist a homeomorphism g : X → g(X) ⊆ I N . Let F is a continuous function from the cantor set C onto
I N , which can be formed using the Lemma-3.1.2. Then F −1 (g(X)) is a closed subset of
C. Again, F F −1 (g(X)) = g(X).
Now X is homeomorphic to g(X) ⊆ I N by the Lemma-3.1.1. Hence X is a continuous
image of the cantor set.
Some important results and definitions
Banach-Mazur Theorem
4. Application
There are several application of the previous theorem. We discuss here only one which
is known as Banach-Mazur theorem.
4.1
Some important results and definitions
Let K be a compact metric space, and denoted by C(k) the Banach space of all continuous
real valued function on K with the supremum norm 1
Definition 4.1.1 — Linearly isometric. A Banach space X is said to be linearly
isometric to a subspace of a Banach space Y if there is a linear operator T : X → y
such that kTx kY = kxkX for every x ∈ X.
Definition 4.1.2 — Linear Topological Vector Space. Let X be a linear space
endowed with a topology T . We call T is a linear topology on X if the following
two operation
+ : X ×X → X
◦ : K×X → X
are continuous. Then (X, T ) is called linear topological vector space.
Note: Every normed linear space is a topological vector space.
Definition 4.1.3 — Weak* Topology. The weak* topology σ (X ∗ , X) is the weakest
topology such that each map
F : X ∗ → K,
continuous.
1 kxk
= sup {|x(t)| : t ∈ K} , x ∈ C(K)
Chapter 4. Application
12
Theorem 4.1.1 — Banach-Alaoglu Theorem. Let X be a Banach space and BX ∗ a
is the unit ball of the dual space X ∗ of X. Then BX ∗ is compact in σ (X ∗ , X) topology.
aB ∗
X
= {x∗ ∈ X ∗ : kx∗ k ≤ 1}
Definition 4.1.4 — metrizable. If X is a topological space, X is said to be metrizable
if there exist a metric d on the set X that induces the topology of X. A metric space is
a metrizable space X together with a specific metric d that gives the topology of X
Theorem 4.1.2 Let X be a Banach space and X ∗ be its dual space. Then X is separable
if and only if BX ∗ is weak* metrizable; that is metrizable in the σ (X ∗ , X) topology.
Corollary 4.1.3 Let X be a separable Banach space. Then the unit ball BX ∗ of X ∗ is
weak* compact and metrizable.
Corollary 4.1.4 Let K be a convex, compact, and metrizable subset of a linear
topological vector space V . Then there is a continuous surjective map from [0, 1] onto
K. More generally, if K is not assumed to be convex, then there is a continuous from
[0, 1] into V whose image contains K
This Corollary is a special case of the Hahn-Mazurkiewicz Theorem, which characterized
the continuous images of the interval [0, 1] as the connected and locally connected
compact metric spaces.
4.2
Banach-Mazur Theorem
Theorem 4.2.1 — Banach-Mazur Theorem. Every separable Banach space is lin-
early isometric to a subspace of C[0, 1]
Proof. The proof has two part:
Part-1: Every separable Banach space is linearly isometric to a subspace of C(K) for some
convex, compact, and metrizable subset K of a linear topological vector space.
Part-2: C(K) is linearly isometric to a subapace of C[0, 1].
We now prove these two part as two Lemmas. The Alexandroff-Hausdroff theorem is
used in second part.:
Lemma 4.2.2 — Part-1. Every separable Banach space is linearly isometric to a sub-
space of C(K) for some convex, compact, and metrizable subset K of a linear topological
vector space.
Proof. Let X is a separable Banach Space, and let X ∗ be its dual. Every element x ∈ X
can be considered to be a function Jx : X ∗ → K given by
Jx (x∗ ) = x∗ (x),
∀x∗ ∈ X
(4.1)
4.2 Banach-Mazur Theorem
13
Of the several topologies that make X ∗ into a linear topological vector space, we use
σ (X ∗ , X)(or weak*) topology.
Now from the definition-4.1.3 Jx are continuous for all x ∈ X. Again B∗X is compact in
weak* topology by Banach-Aloglu Theorem. Since X is also separable so by corollary4.1.3, B∗X is metrizable with respect to weak* topology. Let us choose K = B∗X . Then K
is convex,compact and metrizable in the weak* topology. Using this K we now define
an isometry J of X into C(K) by
(J)(x) = Jx (k) = k(x)
for every x ∈ X and k ∈ K.
Clearly J(x) is continuous on K for each x. The operator J is linear as k ∈ K is linear.
Now for each k ∈ K and x ∈ X
|Jx (k)| = |k(x)| ≤ kkkX ∗ kxkX ≤ kxkX
where the first inequality from the definition of the norm on X ∗ , and second inequality
from the fact that kkkX ∗ ≤ 1 for k in the unit ball K of X ∗ . It follows that
kJ(x)kC(K) = sup {|Jx (k)| : k ∈ K} ≤ kxkX ,
for every x ∈ X.
The reverse inequality follows from the Hahn-Banach Theorem: For every x ∈ X there
exist a point kx ∈ K such that kx (x) = kxkX . It follows that
kJ(x)kC(K) ≥ kJx (kx )k = kkx (x)k = kxkX
Hence the proof.
Lemma 4.2.3 — Part-2. C(K) is linearly isometric to a subapace of C[0, 1].
Proof. Since K is a convex, compact, and metrizable space, the Alxendroff-Hausdorff
Theorem and Corollary-4.1.4 yields a continuous surjective map Φ : [0, 1] → K. The
operator S : C(K) → C[0, 1] is given by
S f (t) = f (φ (t))
for every t ∈ [0, 1] and for some f ∈ C(K).
is a linaer operator from C(K) into C[0, 1], and it is an isometry because
kS f kC[0,1] = sup {| f (φ (t))| : t ∈ [0, 1]} = sup {| f (k)| : k ∈ K} = k f kC(K) ,
where the second equality follows from the surjectivity of φ
5. Conclusion
The Hausdorff - Alexandroff theorem is an important result of theory of compact spaces.
It found its application not only in topology but also in other branches of mathematics
such as geometry and functional analysis. According to Y. Benyamini, it is even possible
to find a common method behind all applications of the Hausdorff-Alexandroff Theorem,
which is given by a possibility to represent any compact metric space in a similar way as
the Cantor set.
Bibliography
Books
[1] James R. Munkres, Topology, PHI Learning Private Limited, 2nd edition, 2013.
[2] W. Rudin, Functional Analysis, McGraw-Hill, 2nd edition, 1991.
[3] J. Hocking and G. Young, Topology, Addison-Wesley, PP. 127-128, Theorem- 3.28.,
1961.
[4] S. Willard, General Topology, Addison-Wesley, PP. 216-218, Theorem- 30.7. 1968.
[5] Frank Burk, LEBESGUE MEASURE AND INTEGRATION- AN INTRODUCTION
A Wiley-Interscience, Appendix A, pp. 252-265, 1998.
[6] Terry J. Morrison, FUNCTIONAL ANALYSIS: An Introduction to Banach Space
Theory, A Wiley-Interscience, pp. 124-142, 2001.
[7] John G. Hocking and Gail S. Young, TOPOLOGY, Dover Publication, Theorem3.28, pp. 127-128, 1961.
Articles
[8] S. Kumeresan, Every compact metric space is the continuous image of the cantor
set, MTTS, PP. 1-3, 2010.
[9] Y. Benyamini, Application of the universal surjectivity of the Cantor set, American
Mathematical Society, Monthly,105, PP. 832-839, 1998.
[10] L. Koudela The Hausdorff-Alxendroff Theorem and its Application in Theory of
Curves, MATFYZ PRESS, Part I, PP. 257-260, 2007.