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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.