DOCSLIB.ORG

Set Known As the Cantor Set. Consider the Clos

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Set Known As the Cantor Set. Consider the Clos COUNTABLE PRODUCTS ELENA GUREVICH Abstract. In this paper, we extend our study to countably infinite products of topological spaces. 1. The Cantor Set Let us constract a very curios (but usefull) set known as the Cantor Set. Consider the 1 2 closed unit interval [0; 1] and delete from it the open interval ( 3 ; 3 ) and denote the remaining closed set by 1 2 G = [0; ] [ [ ; 1] 1 3 3 1 2 7 8 Next, delete from G1 the open intervals ( 9 ; 9 ) and ( 9 ; 9 ), and denote the remaining closed set by 1 2 1 2 7 8 G = [0; ] [ [ ; ] [ [ ; ] [ [ ; 1] 2 9 9 3 3 9 9 If we continue in this way, at each stage deleting the open middle third of each closed interval remaining from the previous stage we obtain a descending sequence of closed sets G1 ⊃ G2 ⊃ G3 ⊃ · · · ⊃ Gn ⊃ ::: T1 The Cantor Set G is defined by G = n=1 Gn, and being the intersection of closed sets, is a closed subset of [0; 1]. As [0; 1] is compact, the Cantor Space (G; τ) ,(that is, G with the subspace topology), is compact. It is useful to represent the Cantor Set in terms of real numbers written to basis 3, that is, 5 ternaries. In the ternary system, 76 81 would be written as 2211:0012, since this represents 2 · 33 + 2 · 32 + 1 · 31 + 1 · 30 + 0 · 3−1 + 0 · 3−2 + 1 · 3−3 + 2 · 3−4 So a number x 2 [0; 1] is represented by the base 3 number a1a2a3 : : : an ::: where 1 X an x = a 2 f0; 1; 2g 8n 2 3n n N n=1 Turning again to the Cantor Set G, it should be clear that an element of [0; 1] is in G if 1 5 and only if it can be written in ternary form with an 6= 1 8n 2 N, so 2 2= G 81 2= G but 1 1 1 3 2 G and 1 2 G. [we denote 2 = 0:1111 ::: , 3 = 0:02222 ::: 1 = 0:2222 ::: ] Thus we have a function f from the Cantor Set into the set of all sequences of the form < a1; a2; a3; : : : ; an; ··· >, where each ai 2 f0; 2g and f is one-to-one and onto. 2. The Product Topology Definition 2.1. Let (X1; τ1); (X2; τ2);:::; (Xn; τn);::: be a countably infinite family of Q1 topological spaces. Then the product i=1 Xi of the sets Xi, i 2 N consists of all the infinite sequences < x1; x2; x3; : : : ; xn; ··· > where xi 2 Xi for all i. The product space Date: May 6, 2011. 1 2 Q1 Q1 i=1(Xi; τi) consists of the product i=1 Xi with the topology τ having as its basis the family 1 Y 0 B = f Oi : Oi 2 τi; and Oi = Xi for all but a finite number of i sg i=1 0 Proposition 2.2. Let (Xi; τi), (Yi; τi ) i 2 N, be countably infinite families of topological Q1 Q1 0 spaces, having product spaces ( i=1 Xi; τ)( i=1 Yi; τ ) respectively. If the mapping hi : 0 Q1 (Xi; τi) ! (Yi; τi ) is continuous for each i 2 N then so is the mapping - h :( i=1 Xi; τ) ! Q1 0 ( i=1 Yi; τ ) given by h(< x1; x2; x3; : : : ; xn; ··· >) =< h1(x1); h2(x2); h3(x3); : : : ; hn(xn); ··· > Q1 −1 Proof. It suffices to show that if O is a basic open set in ( i=1 Xi; τ) then h (O) is open Q1 0 0 in ( i=1 Yi; τ ). Consider the basic open set U1 × U2 × · · · × Un × Yn+1 × ::: where Ui 2 τi for i = 1; 2; : : : ; n. Then −1 −1 −1 −1 h (U1 × U2 × · · · × Un × Yn+1 × ::: ) = h (U1) × h (U2) × · · · × h (Un) × Xn+1 × ::: −1 since the continuity of each hi implies that hi (ui) 2 τi for i = 1; 2; : : : ; n. So h is continu- ous. 3. The Cantor space and the Hilbert cube Proposition 3.1. Let (G; τ) be the Cantor space and for each i 2 N let (Ai; τi) be the Q1 set 0; 2 with the discrete topology, and let ( i=1 Ai; τ) be it's product space. Then the map 0 Q1 P1 an f :(G; τ) ! ( i=1 Ai; τ ) given by f( i=1 3n ) =< a1; a2; a3; : : : ; an; ··· > is a homeomor- phism. Proof. It is easy to see that f is one-to-one and onto. To prove that f is continuous it suffices to show for any basic given set U = U1 × U2 × · · · × Un × An+1 × ::: and any P1 an point a =< a1; a2; a3; : : : ; an; ··· >2 U there exists an open set W 3 i=1 3n such that P1 an 1 P1 an 1 f(W ) ⊆ U. Consider the open interval ( i=1 3n − 3N+2 ; i=1 3n + 3N+2 ) and let W be the P xn intersection of this open intervals with G. Then W is open in (G; τ) and if x = 3n 2 W then xi = ai for i = 1; 2;:::;N. So f(x) 2 U1 × U2 × · · · × Un × An+1 × ::: and thus f(W ) ⊆ U as required. Proposition 3.2. Let (Gi; τi), i 2 N, be a countably infinite family of topological spaces Q1 each of which is homeomorphic to the Cantor space (G; τ) then (G; τ) ' i=1(Gi; τi) ' Qn i=1(Gi; τi) for each n 2 N. Proof. First let us verify that (G; τ) ' (G1; τ1)×(G2; τ2). This is, by proposition 3.1 equiva- Q1 Q1 Q1 lent to showing that i=1(Ai; τi) ' i=1(Ai; τi)× i=1(Ai; τi) where each (Ai; τi) is the set Q1 Q1 0; 2 with the discrete topology. Now we define a function Θ : i=1(Ai; τi) × i=1(Ai; τi) ! Q1 i=1(Ai; τi) by Θ(< a1; a2; a3; ··· >; < b1; b2; b3; ··· >) =< a1; b1; a2; b2; ··· >. It's easy to show that Θ is a homeomorphism, and so (G1; τ1) × (G2; τ2 ' (G; τ). By induction we get Q1 that (G; τ) ' i=1(Ai; τi) for every positive integer n. To show the infinite product case, define the mapping 1 1 1 Y Y Y Φ:[ (Ai; τi) × (Ai; τi) × ::: ] ! (Ai; τi) i=1 i=1 i=1 by Φ(< a1; a2; a3; ··· >; < b1; b2; b3; ··· >; < c1; c2; c3; ··· >; < d1; d2; d3; ··· >; < e1; e2; e3; ··· > ) =< a1; a2; b1; a3; b2; c1; a4; b3; c2; d1; a5; b4; c3; d2; e1; ··· > Again it is easily verified that Φ is homeomorphism and the proof is complete. 3 Proposition 3.3. The topological space [0; 1] is a continuous image of the Cantor space (G; τ). Q1 Proof. It suffices to find a continuous mapping Φ of i=1(Ai; τi) onto [0; 1]. Such a mapping is given by 1 X ai Φ(< a ; a ; : : : ; a ; ··· >) = 1 2 i 2i+1 i=1 It is easy to show that Φ is onto and continuous, and the proof is complete. Definition 3.4. For each positive integer n, let the topological space (In; τn) be homeomor- Q1 phic to [0; 1]. Then the product space n=1(In; τn) is called Hilbert cube and is denoted 1 Qn n by I . The product space i=1(Ii; τi) is called the n-cube, and is denoted by I , for each n 2 N Theorem 3.5. The Hilbert cube is compact. 0 Proof. By proposition 3.3 there is a continuous mapping Φn of (Gn; τn) onto (In; τn where 0 for each n 2 N,(Gn; τn) and (In; τn are homeomorphic to the Cantor space and [0; 1] respec- Q1 Q1 0 1 tively. Therefore, there is a continuous mapping Ψ of n=1(Gn; τn) onto n=1(In; τn = I Q1 1 but i=1(Gn; τn) is homeomorphic to the Cantor space (G; τ). Therefore I is a continuous image of the compact space (G; τ) and hance is compact. Proposition 3.6. Let (Xi; τi) i 2 N be a countably infinite family of matrizable spaces, Q1 then i=1(Xi; τi) is metrizable. Corollary 3.7. The Hilbert cube is metrizable. 4. The Cantor space and the Hilbert cube Definition 4.1. A topological space (X; τ) is said to be separable if it has a countable dense subset. Proposition 4.2. Let (X; τ) be a compact metrizable space. Then (X; τ) is separable. Proof. Let d be a metric on X which induces the topology jtau. For each positive integer n, 1 let Sn be the family of all open balls having centers in X and radius n .Then Sn is an open covering of X and so there is a finite sub covering Un = fUn1 ;Un2 ;:::;Unk g for some k 2 N. 1 Let ynj be the center of Un, j = 1; 2; : : : ; k and YN = fyn1 ; : : : ; ynk g. Put Y = [n=1Yn, then Y is a countable subset of X. Now, if V is any non open set in (X; τ), then for any v 2 V , 1 V contains an open ball B of radius n . As Un is an open cover of X, v 2 Unj for some j. 1 Thus d(v; ynj ) < n , and so ynj 2 B ⊂ V . Hance V \ Y 6= ? and so Y is dense in X. Corollary 4.3. The Hilbert cube is separable space. Definition 4.4. A topological space (X; τ) is said to be a T1space if every singelton set fxg, x 2 X, is a closed set.
Load more

Recommended publications
  • Martin-L¨Of Random Points Satisfy Birkhoff's Ergodic
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000{000 S 0002-9939(XX)0000-0 MARTIN-LOF¨ RANDOM POINTS SATISFY BIRKHOFF'S ERGODIC THEOREM FOR EFFECTIVELY CLOSED SETS JOHANNA N.Y. FRANKLIN, NOAM GREENBERG, JOSEPH S. MILLER, AND KENG MENG NG Abstract. We show that if a point in a computable probability space X sat- isfies the ergodic recurrence property for a computable measure-preserving T : X ! X with respect to effectively closed sets, then it also satisfies Birkhoff's ergodic theorem for T with respect to effectively closed sets. As a corollary, every Martin-L¨ofrandom sequence in the Cantor space satisfies 0 Birkhoff's ergodic theorem for the shift operator with respect to Π1 classes. This answers a question of Hoyrup and Rojas. Several theorems in ergodic theory state that almost all points in a probability space behave in a regular fashion with respect to an ergodic transformation of the space. For example, if T : X ! X is ergodic,1 then almost all points in X recur in a set of positive measure: Theorem 1 (See [5]). Let (X; µ) be a probability space, and let T : X ! X be ergodic. For all E ⊆ X of positive measure, for almost all x 2 X, T n(x) 2 E for infinitely many n. Recent investigations in the area of algorithmic randomness relate the hierarchy of notions of randomness to the satisfaction of computable instances of ergodic theorems. This has been inspired by Kuˇcera'sclassic result characterising Martin- L¨ofrandomness in the Cantor space. We reformulate Kuˇcera'sresult using the general terminology of [4].
    [Show full text]
  • The Homogeneous Property of the Hilbert Cube 3
    THE HOMOGENEOUS PROPERTY OF THE HILBERT CUBE DENISE M. HALVERSON AND DAVID G. WRIGHT Abstract. We demonstrate the homogeneity of the Hilbert Cube. In partic- ular, we construct explicit self-homeomorphisms of the Hilbert cube so that given any two points, a homeomorphism moving one to the other may be realized. 1. Introduction It is well-known that the Hilbert cube is homogeneous, but proofs such as those in Chapman’s CBMS Lecture Notes [1] show the existence of a homeomorphism rather than an explicit homeomorphism. The purpose of this paper is to demonstrate how to construct explicit self-homeomorphisms of the Hilbert cube so that given any two points, a homeomorphism moving one to the other may be realized. This is a remarkable fact because in a finite dimensional manifold with boundary, interior points are topologically distinct from boundary points. Hence, the n-dimensional cube n n C = Π Ii, i=1 where Ii = [−1, 1], fails to be homogeneous because there is no homeomorphism of Cn onto itself mapping a boundary point to an interior point and vice versa. Thus, the fact that the Hilbert cube is homogeneous implies that there is no topological distinction between what we would naturally define to be the boundary points and the interior points, based on the product structure. Formally, the Hilbert cube is defined as ∞ Q = Π Ii i=1 where each Ii is the closed interval [−1, 1] and Q has the product topology. A point arXiv:1211.1363v1 [math.GT] 6 Nov 2012 p ∈ Q will be represented as p = (pi) where pi ∈ Ii.
    [Show full text]
  • Topological Properties of the Hilbert Cube and the Infinite Product of Open Intervals
    TOPOLOGICAL PROPERTIES OF THE HILBERT CUBE AND THE INFINITE PRODUCT OF OPEN INTERVALS BY R. D. ANDERSON 1. For each i>0, let 7, denote the closed interval O^x^ 1 and let °I¡ denote the open interval 0<*< 1. Let 7°°=fli>o 7 and °7°°= rii>o °h- 7°° is the Hilbert cube or parallelotope sometimes denoted by Q. °7°° is homeomorphic to the space sometimes called s, the countable infinite product of lines. The principal theorems of this paper are found in §§5, 7, 8, and 9. In §5 it is shown as a special case of a somewhat more general theorem that for any countable set G of compact subsets of °7°°, °7eo\G* is homeomorphic to °7°° (where G* denotes the union of the elements of G)(1). In §7, it is shown that a great many homeomorphisms of closed subsets of 7" into 7" can be extended to homeomorphisms of 7™ onto itself. The conditions are in terms of the way in which the sets are coordinatewise imbedded in 7°°. A corol- lary is the known fact (Keller [6], Klee [7], and Fort [5]) that if jf is a countable closed subset of 7°°, then every homeomorphism of A"into 7 e0can be extended to a homeomorphism of 7°° onto itself. In a further paper based on the results and methods of this paper, the author will give a topological characterization in terms of imbeddings of those closed subsets X of 7 e0 for which homeomorphisms of X into Wx={p \pelx and the first coordinate of p is zero} can be extended to homeomorphisms of X onto itself.
    [Show full text]
  • 2' Is Homeomorphic to the Hilbert Cube1
    BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 78, Number 3, May 1972 2' IS HOMEOMORPHIC TO THE HILBERT CUBE1 BY R. SCHORI AND J. E. WEST Communicated by Steve Armentrout, November 22, 1971 1. Introduction. For a compact metric space X, let 2X be the space of all nonempty closed subsets of X whose topology is induced by the Hausdorff metric. One of the well-known unsolved problems in set-theoretic topology has been to identify the space 21 (for I = [0,1]) in terms of a more man­ ageable definition. Professor Kuratowski has informed us that the con­ jecture that 21 is homeomorphic to the Hilbert cube Q was well known to the Polish topologists in the 1920's. In 1938 in [7] Wojdyslawski specifically asked if 2* « Q and, more generally, he asked if 2X « Q where X is any nondegenerate Peano space. In this paper we outline our rather lengthy proof that 21 « Q, announce some generalizations to some other 1- dimensional X9 and give some of the technical details. 2. Preliminaries. If X is a compact metric space, then the Hausdorff metric D on 2X can be defined as D(A, B) = inf{e :A c U(B, e) and B a U(A9 e)} where, for C c X, U(C9 s) is the e-neighborhood of C in X. An inverse sequence (Xn, ƒ„) will have, for n^l, bonding maps fn:Xn + l -• X„ and the inverse limit space will be denoted by lim(ZM, ƒ„). The theory of near-homeomorphisms is very important in this work (see §5).
    [Show full text]
  • Characterization of Cantor Spaces
    Characterization of Cantor Spaces Matthew Shaw (ref. Pugh Analysis Textbook) November 2019 1 Introduction The standard Cantor Middle Thirds Set is compact, perfect, nonempty, and to- tally disconnected (and since the construction takes place in R with its standard metric, the Cantor Set is also metrizable). Totally Disconnected: Every connected component is a singleton set, and equiv- alently, every point has arbitrarily small clopen neighborhoods. Perfect: A space is called perfect if it has no isolated points. There are two main theorems of this presentation: The Cantor Surjection The- orem, that every nonempty compact metric space is the image of a continuous function with the Cantor set as its domain, and a complete characterization of Cantor spaces, in particular that every compact, nonempty, perfect, and to- tally disconnected metric space is homeomorphic to the standard Cantor middle thirds set. Notation: The set of words of countably infinite length in a two character al- phabet is denoted by Ω, the standard Cantor Middle-Thirds set is denoted by C, and the set of words of length n (where n 2 N) in a two character alphabet is denoted !(n). Note that the size of !(n) is always 2n. ! with no associated number will represent a word of countably infinite length (i.e. an element of Ω) and the notation !jn for n 2 N means the word ! truncated to only its first n characters (this is an element of !(n)). t denotes disjoint union of sets. If α and β are words of finite length, then the notation αβ means the word made up of the characters of α followed by the characters of β, which is also termed concatenation.
    [Show full text]
  • Hyperspaces of Compact Convex Sets
    Pacific Journal of Mathematics HYPERSPACES OF COMPACT CONVEX SETS SAM BERNARD NADLER,JR., JOSEPH E. QUINN AND N. STAVRAKAS Vol. 83, No. 2 April 1979 PACIFIC JOURNAL OF MATHEMATICS Vol. 83, No. 2, 1979 HYPERSPACES OF COMPACT CONVEX SETS SAM B. NADLER, JR., J. QUINN, AND NICK M. STAVRAKAS The purpose of this paper is to develop in detail certain aspects of the space of nonempty compact convex subsets of a subset X (denoted cc(X)) of a metric locally convex T V.S. It is shown that if X is compact and dim (X)^2then cc(X) is homeomorphic with the Hubert cube (denoted o,c{X)~IJ). It is shown that if w^2, then cc(Rn) is homeomorphic to 1^ with a point removed. More specialized results are that if XaR2 is such that ccCX)^ then X is a two cell; and that if XczRz is such that ccCX)^/^ and X is not contained in a hyperplane then X must contain a three cell. For the most part we will be restricting ourselves to compact spaces X although in the last section of the paper, § 7, we consider some fundamental noncompact spaces. We will be using the following definitions and notation. For each n = 1,2, , En will denote Euclidean w-space, Sn~ι = {xeRn: \\x\\ = 1}, Bn = {xeRn: \\x\\ ^ 1}, and °Bn = {xeRn: \\x\\<l}. A continuum is a nonempty, compact, connected metric space. An n-cell is a continuum homeomorphic to Bn. The symbol 1^ denotes the Hilbert cube, i.e., /«, = ΠΓ=i[-l/2*, 1/2*].
    [Show full text]
  • Measures from Infinite-Dimensional Gauge Integration Cyril Levy
    Measures from infinite-dimensional gauge integration Cyril Levy To cite this version: Cyril Levy. Measures from infinite-dimensional gauge integration. 2019. hal-02308813 HAL Id: hal-02308813 https://hal.archives-ouvertes.fr/hal-02308813 Preprint submitted on 8 Oct 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Measures from infinite-dimensional gauge integration Cyril Levy IMT Toulouse – INUC Albi Email address: [email protected] January 2019 Abstract We construct and investigate an integration process for infinite products of compact metrizable spaces that generalizes the standard Henstock–Kurzweil gauge integral. The integral we define here relies on gauge functions that are valued in the set of divisions of the space. We show in particular that this integration theory provides a unified setting for the study of non- T absolute infinite-dimensional integrals such as the gauge integrals on R of Muldowney and the con- struction of several type of measures, such as the Lebesgue measure, the Gaussian measures on Hilbert spaces, the Wiener measure, or the Haar measure on the infinite dimensional torus. Furthermore, we characterize Lebesgue-integrable functions for those measures as measurable absolutely gauge inte- grable functions.
    [Show full text]
  • Does Minimal Action by the Group of Measure Preserving Transformations Imply the Existence of a Uniquely Ergodic Transformation for a Measure on a Cantor Set?
    DOES MINIMAL ACTION BY THE GROUP OF MEASURE PRESERVING TRANSFORMATIONS IMPLY THE EXISTENCE OF A UNIQUELY ERGODIC TRANSFORMATION FOR A MEASURE ON A CANTOR SET? ETHAN AKIN Assume that a Borel probability measure µ on a Cantor set X satisfies µ(A) > 0 if A is opene (=open and nonempty) and µ(A) = 0 if A is countable, ie. µ is full and nonatomic. For such a measure the clopen values set S(µ) =def {µ(A): A a clopen subset of X} is a countable dense subset of the interval [0, 1] which includes the endpoints 0, 1. Such a subset is called grouplike if it is the intersection of [0, 1] with a dense additive subgroup of the reals. For such a measure µ we define for any clopene (= clopen and nonempty) subset A the relative measure µA by µA(B) = µ(B)/µ(A) for measurable B ⊂ A. Thus, µA is a full, nonatomic probability measure on the Cantor set A. Definition 1. A full, nonatomic probability measure µ on a Cantor set X is called good if for all clopen subsets A, B of X with µ(A) < µ(B) there exists a clopen subset A1 of B such that µ(A) = µ(A1). These measures satisfy the following properties, see Akin (2003) Good measures on Cantor space (to appear Trans. AMS). (1) Among good measures the clopen values set provides a complete topological invariant. (2) If µ is a good measure on X then S(µ) is grouplike and every grouplike subset is the clopen values set for some good measure.
    [Show full text]
  • Identifying Hilbert Cubes: General Methods and Their Application to Hyperspaces by Schori and West
    Toposym 3 James E. West Identifying Hilbert cubes: general methods and their application to hyperspaces by Schori and West In: Josef Novák (ed.): General Topology and its Relations to Modern Analysis and Algebra, Proceedings of the Third Prague Topological Symposium, 1971. Academia Publishing House of the Czechoslovak Academy of Sciences, Praha, 1972. pp. 455--461. Persistent URL: http://dml.cz/dmlcz/700806 Terms of use: © Institute of Mathematics AS CR, 1972 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz 455 mENTIFYING HILBERT CUBES: GENERAL METHODS AND THEIR APPLICATION TO HYPERSPACES BY SCHORI AND WEST J. E. WEST Ithaca In the past several years new methods of identifying Hilbert cubes have been discovered, and they have been applied by R. M. Schori and the author to hyper- spaces of some one-dimensional Peano continua. In particular, we have solved affirmatively the conjecture of M. Wojdyslawski [11] that the hyperspace 2* of all non-void, closed subsets of the interval is a Hilbert cube, when topologized by the Hausdorff metric. I am informed by Professor Kuratowski that the conjecture was originally posed in the 1920's and was well-known to topologists ip Warsaw and other places at that time. This report outlines some of these methods and the Schori- West proof of this conjecture.
    [Show full text]
  • Rigid Sets in the Hilbert Cube
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Topology and its Applications 22 (1986) 85-96 85 North-Holland RIGID SETS IN THE HILBERT CUBE Jack W. LAMOREAUX and David G. WRIGHT Department of Mathematics, Brigham Young University, Provo, UT 84602, USA Received 27 November 1984 Revised 29 March 1985 Let X be a compact metric space with no isolated points. Then we may embed X as a subset of the Hilbert cube Q (Xc Q) so that the only homeomorphism of X onto itself that extends to a homeomorphism of Q is the identity homeomorphism. Such an embedding is said to be rigid. In fact, there are uncountably many rigid embeddings X, of X in Q so that for (I f /!I and any homeomorphism h of Q, h(X,) fl X, is a Z-set in Q and a nowhere dense subset of each of h(X,) and X,. AMS (MOS) Subj. Class.: Primary 54C25, 57N35; 1. Introduction Let X be a subset of a space Y. We say that X is rigid in Y if the only homeomorphism of X onto itself that is extendable to a homeomorphism of Y is the identity homeomorphism. Examples of interesting rigid sets in 3-space have been given by Martin [lo], Bothe [4], and Shilepsky [ 141. Shilepsky’s example was based on work of Sher [13]. More recently, Wright [15] has made a study of rigid sets in Euclidean n-space which generalizes the 3-dimensional results mentioned above. The program outlined in [15] leads one to make the following conjecture about the Hilbert cube.
    [Show full text]
  • An Introduction to Totally Disconnected Locally Compact Groups
    An introduction to totally disconnected locally compact groups Phillip R. Wesolek December 11, 2018 ii Contents 1 Topological Structure 7 1.1 van Dantzig's theorem . .7 1.2 Isomorphism theorems . 11 1.3 Graph automorphism groups . 14 1.4 Exercises . 20 2 Haar Measure 23 2.1 Functional analysis . 23 2.2 Existence . 25 2.3 Uniqueness . 33 2.4 The modular function . 37 2.5 Quotient integral formula . 40 2.6 Exercises . 44 3 Geometric Structure 49 3.1 The Cayley{Abels graph . 49 3.2 Cayley-Abels representations . 53 3.3 Uniqueness . 57 3.4 Compact presentation . 60 3.5 Exercises . 62 4 Essentially Chief Series 65 4.1 Graphs revisited . 66 4.2 Chain conditions . 72 4.3 Existence of essentially chief series . 75 4.4 Uniqueness of essentially chief series . 78 4.5 The refinement theorem . 80 4.6 Exercises . 87 1 2 CONTENTS Introduction Preface For G a locally compact group, the connected component which contains the identity, denoted by G◦, is a closed normal subgroup. This observation produces a short exact sequence of topological groups f1g ! G◦ ! G ! G=G◦ ! f1g where G=G◦ is the group of left cosets of G◦ endowed with the quotient topology. The group G◦ is a connected locally compact group, and the group of left cosets G=G◦ is a totally disconnected locally compact (t.d.l.c.) group. The study of locally compact groups therefore in principle, although not always in practice, reduces to studying connected locally compact groups and t.d.l.c. groups. The study of locally compact groups begins with the work [11] of S.
    [Show full text]
  • Countable Metric Spaces Without Isolated Points
    Topology Explained. June 2005. Published by Topology Atlas. Document # paca-25. c 2005 by Topology Atlas. This paper is online at http://at.yorku.ca/p/a/c/a/25.htm. COUNTABLE METRIC SPACES WITHOUT ISOLATED POINTS ABHIJIT DASGUPTA Theorem (Sierpinski, 1914–1915, 1920). Any countable metrizable space without isolated points is homeomorphic to Q, the rationals with the order topology (same as Q as a subspace of R with usual topology, or as Q with the metric topology). The theorem is remarkable, and gives some apparently counter-intuitive exam- ples of spaces homeomorphic to the usual Q. Consider the “Sorgenfrey topology on Q,” which has the collection {(p, q]: p, q ∈ Q} as a base for its topology. This topology on Q is strictly finer than, and yet homeomorphic to, the usual topology of Q. Another example is Q × Q as a subspace of the Euclidean plane. In this article, we present three proofs of Sierpinski’s theorem. 1. Order-theoretic proof This proof is fairly elementary in the sense that no “big guns” are used (such as Brouwer’s characterization of the Cantor space or the Alexandrov-Urysohn charac- terization of the irrationals), and no new back-and-forth method is used, but the main tool is: Theorem (Cantor’s Theorem). Any countable linear order which is order dense (meaning x < y implies there is z such that x < z < y), and has no first or last element is order isomorphic to (Q, <). This theorem will be used more than once. N N 1.1. Easy properties of 2 = Z2 .
    [Show full text]