Topological Properties of the Cantor Spaces
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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]. -
D-18 Theˇcech–Stone Compactifications of N and R
d-18 The Cech–Stoneˇ compactifications of N and R 213 d-18 The Cech–Stoneˇ Compactifications of N and R c It is safe to say that among the Cech–Stoneˇ compactifica- The proof that βN has cardinality 2 employs an in- tions of individual spaces, that of the space N of natural dependent family, which we define on the countable set numbers is the most widely studied. A good candidate for {hn, si: n ∈ N, s ⊆ P(n)}. For every subset x of N put second place is βR. This article highlights some of the most Ix = {hn, si: x ∩ n ∈ s}. Now if F and G are finite dis- striking properties of both compactifications. T S joint subsets of P(N) then x∈F Ix \ y∈G Iy is non-empty – this is what independent means. Sending u ∈ βN to the P( ) ∗ point pu ∈ N 2, defined by pu(x) = 1 iff Ix ∈ u, gives us a 1. Description of βN and N c continuous map from βN onto 2. Thus the existence of an independent family of size c easily implies the well-known The space β (aka βω) appeared anonymously in [10] as an N fact that the Cantor cube c2 is separable; conversely, if D is example of a compact Hausdorff space without non-trivial c dense in 2 then setting Iα = {d ∈ D: dα = 0} defines an converging sequences. The construction went as follows: for ∗ ∈ independent family on D. Any closed subset F of N such every x (0, 1) let 0.a1(x)a2(x).. -
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. -
Absolute Measurable Spaces ENCYCLOPEDIAOF MATHEMATICSAND ITS APPLICATIONS
A bsolute M easurable S paces Absolutemeasurablespaceandabsolutenull spaceareveryold topological notions, developed from descriptive set theory, topology, Borel measure theory and analysis. This monograph systematically develops and returns to the topological and geometrical origins of these notions. Motivating the development of the exposition aretheaction of thegroup of homeomorphismsof a spaceon Borelmeasures,the Oxtoby–Ulam theorem on Lebesgue-like measures on the unit cube, and the extensions of this theorem to many other topological spaces. Existence of uncountable absolute null space, extension of the Purves theorem, and recent advances on homeomorphic Borel probability measures on the Cantor space are among the many topics discussed. A brief discussion of set-theoretic results on absolutenull spaceis also given. A four-part appendix aids the reader with topological dimension theory, Hausdorff measure and Hausdorff dimension, and geometric measure theory. The exposition will suit researchers and graduate students of real analysis, set theory and measure theory. Togo Nishiurais Professor Emeritus at Wayne State University, Detroit, andAsso- ciate Fell in Mathematics at Dickinson College, Pennsylvania. ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS All the titles listed bel can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit http://www.cambridge.org/uk/series/sSeries.asp?code=EOM! 63. A. C. Thompson Minkowski Geometry 64 R. B. Bapat and T. E. S. Raghavan Nonnegative Matrices with Applications 65. K. Engel Sperner Theory 66. D. Cvetkovic, P. Rlinson, and S. Simic Eigenspaces of Graphs 67. F. Bergeron, G. Labelle, and P. Leroux Combinational Species and Tree-Like Structures 68. R. Goodman and N. -
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. -
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. -
Dense Subgroups of Compact Groups
Open Problems in Topology II – Edited by E. Pearl © 2007 Elsevier B.V. All rights reserved Dense subgroups of compact groups W. W. Comfort 1. Introduction The symbol G here denotes the class of all infinite groups, and TG denotes the class of all infinite topological groups which satisfy the T0 separation prop- erty. Each element of TG is, then, a Tychonoff space (i.e., a completely regular, Hausdorff space) [50, 8.4]. We say that G = (G, T ) ∈ TG is totally bounded (some authors prefer the expression pre-compact) if for every U ∈ T \ {∅} there is a finite set F ⊆ G such that G = FU. Our point of departure is the following portion of Weil’s Theorem [77]: Every totally bounded G ∈ TG embeds as a dense topological subgroup of a compact group G; this is unique in the sense that for every compact group Ge containing G densely there is a homeomorphism-and- isomorphism ψ : G Ge fixing G pointwise. As usual, a space is ω-bounded if each of its countable subsets has compact closure. For G = (G, T ) ∈ TG we write G ∈ C [resp., G ∈ Ω; G ∈ CC; G ∈ P; G ∈ TB] if (G, T ) is compact [resp., ω-bounded; countably compact; pseudocompact; totally bounded]. And for X ∈ {C, Ω, CC, P, TB} and G ∈ G we write G ∈ X0 if G admits a group topology T such that (G, T ) ∈ X. The class-theoretic inclusions C ⊆ Ω ⊆ CC ⊆ P ⊆ TB ⊆ TG and C0 ⊆ Ω0 ⊆ CC0 ⊆ P0 ⊆ TB0 ⊆ TG0 = G are easily established. (For P ⊆ TB, see [31]. -
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. -
Topology Proceedings
Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124 COPYRIGHT °c by Topology Proceedings. All rights reserved. TOPOLOGY PROCEEDINGS Volume 27, No. 2, 2003 Pages 545{558 THE TOPOLOGY OF COMPACT GROUPS SIDNEY A. MORRIS∗ Abstract. This paper is a slightly expanded version of a plenary ad- dresss at the 2002 Summer Topology Conference in Auckland, New Zealand. The aim in this paper is not to survey the vast literature on compact groups, or even the topological group structure of compact groups which is described in considerable detail in the 1998 book \The Structure of Compact Groups: A Primer for the Student { A Handbook for the Expert" by Karl Heinrich Hofmann and this author. Rather, the much more modest purpose in this article is to focus on point-set topology and, in a gentle fashion, describe the topological structure of compact groups, most of which can be extracted or derived from that large book. 1. Fundamental Facts Portions of the material presented here can be found in various papers and books. The book ([3],[4]) by Karl Heinrich Hofmann and the author is a comfortable reference. However, this is not meant to imply that all the results in [3] were first proved by Hofmann and/or Morris, although often the presentation and approach in the book are new. Definition 1.1. Let G be a group with a topology τ. Then G is said to be a topological group if the maps G G given by g g 1 and G G G ! 7! − × ! given by (g1; g2) g1:g2 are continuous. -
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 . -
Formal Topology and Constructive Compactness of the Cantor Space
Formal topology and constructive compactness of the Cantor space Ben Sherman November 11, 2016 1 Introduction Many mathematical concepts involve structures that are best understood not as discrete sets of points but rather as something more continuous, more akin to a connected and indivisible mass. For instance, geometry, analysis, probability, all rely on notions such as time and space, and distance and measure, which have this character. Topology studies how to make sense of these spaces. The \classical" theory of general topology, which defines topological spaces in terms of the basic notion of open sets, can be traced back to Felix Hausdorff's definition of neighborhood spaces in 1914. Hausdorff's definition followed a series of other attempts at axiomatizing general topology in terms of other notions such as limits, accumulation points, and metrics. Notably, all of these definitions took as given that a space should be defined as a set of points with some additional properties. Brouwer, in his 1913 Intuitionism and Formalism, denies that a continuum should be identified as a set of points, declaring in his \second act of intuitionism" that \the linear continuum... is not exhaustible by the interposition of new units and therefore can never be thought of as a mere collection of units." Moreover, Brouwer introduced a notion of choice sequences. Consider the space of sequences of boolean values, N ! B. Brouwer denied that every such sequence σ : N ! B had to be necessarily determined by a law which specified the sequence in its entirety in advance. Rather, he envisioned σ as exposing an interaction structure. -
A B S T R a C T. We Show That There Is a Scattered Compact Subset K of The
Bulletin T.CXXXI de l’Acad´emieserbe des sciences et des arts ¡ 2005 Classe des Sciences math´ematiqueset naturelles Sciences math´ematiques, No 30 REPRESENTING TREES AS RELATIVELY COMPACT SUBSETS OF THE FIRST BAIRE CLASS S. TODORCEVIˇ C´ (Presented at the 7th Meeting, held on October 29, 2004) A b s t r a c t. We show that there is a scattered compact subset K of the first Baire class, a Baire space X and a separately continuous mapping f : X £ K ¡! R which is not continuous on any set of the form G £ K, where G is a comeager subset of X. We also show that it is possible to have a scattered compact subset K of the first Baire class which does have the Namioka property though its function space C(K) fails to have an equivalent Fr´echet-differentiable norm and its weak topology fails to be σ-fragmented by the norm. AMS Mathematics Subject Classification (2000): 46B03, 46B05 Key Words: Baire Class-1, Function spaces, Renorming Recall the well-known classical theorem of R.Baire which says that for ev- ery separately continuous real function defined on the unit square [0; 1]£[0; 1] is continuous at every point of a set of the form G£[0; 1] where G is a comea- ger subset of [0; 1]. This result was considerably extended by I.Namioka [12] who showed that the same conclusion can be reached for separately contin- uous functions on products of the form X £ K where X is a Cech-complete space and where K is a compact space.