Countable Metric Spaces Without Isolated Points
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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]. -
12 | Urysohn Metrization Theorem
12 | Urysohn Metrization Theorem In this chapter we return to the problem of determining which topological spaces are metrizable i.e. can be equipped with a metric which is compatible with their topology. We have seen already that any metrizable space must be normal, but that not every normal space is metrizable. We will show, however, that if a normal space space satisfies one extra condition then it is metrizable. Recall that a space X is second countable if it has a countable basis. We have: 12.1 Urysohn Metrization Theorem. Every second countable normal space is metrizable. The main idea of the proof is to show that any space as in the theorem can be identified with a subspace of some metric space. To make this more precise we need the following: 12.2 Definition. A continuous function i: X → Y is an embedding if its restriction i: X → i(X) is a homeomorphism (where i(X) has the topology of a subspace of Y ). 12.3 Example. The function i: (0; 1) → R given by i(x) = x is an embedding. The function j : (0; 1) → R given by j(x) = 2x is another embedding of the interval (0; 1) into R. 12.4 Note. 1) If j : X → Y is an embedding then j must be 1-1. 2) Not every continuous 1-1 function is an embedding. For example, take N = {0; 1; 2;::: } with the discrete topology, and let f : N → R be given ( n f n 0 if = 0 ( ) = 1 n if n > 0 75 12. -
General Topology
General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry). -
An Exploration of the Metrizability of Topological Spaces
AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES DUSTIN HEDMARK Abstract. A study of the conditions under which a topological space is metrizable, concluding with a proof of the Nagata Smirnov Metrization The- orem Contents 1. Introduction 1 2. Product Topology 2 3. More Product Topology and R! 3 4. Separation Axioms 4 5. Urysohn's Lemma 5 6. Urysohn's Metrization Theorem 6 7. Local Finiteness and Gδ sets 8 8. Nagata-Smirnov Metrization Theorem 11 References 13 1. Introduction In this paper we will be exploring a basic topological notion known as metriz- ability, or whether or not a given topology can be understood through a distance function. We first give the reader some basic definitions. As an outline, we will be using these notions to first prove Urysohn's lemma, which we then use to prove Urysohn's metrization theorem, and we culminate by proving the Nagata Smirnov Metrization Theorem. Definition 1.1. Let X be a topological space. The collection of subsets B ⊂ X forms a basis for X if for any open U ⊂ X can be written as the union of elements of B Definition 1.2. Let X be a set. Let B ⊂ X be a collection of subsets of X. The topology generated by B is the intersection of all topologies on X containing B. Definition 1.3. Let X and Y be topological spaces. A map f : X ! Y . If f is a homeomorphism if it is a continuous bijection with a continuous inverse. If there is a homeomorphism from X to Y , we say that X and Y are homeo- morphic. -
DEFINITIONS and THEOREMS in GENERAL TOPOLOGY 1. Basic
DEFINITIONS AND THEOREMS IN GENERAL TOPOLOGY 1. Basic definitions. A topology on a set X is defined by a family O of subsets of X, the open sets of the topology, satisfying the axioms: (i) ; and X are in O; (ii) the intersection of finitely many sets in O is in O; (iii) arbitrary unions of sets in O are in O. Alternatively, a topology may be defined by the neighborhoods U(p) of an arbitrary point p 2 X, where p 2 U(p) and, in addition: (i) If U1;U2 are neighborhoods of p, there exists U3 neighborhood of p, such that U3 ⊂ U1 \ U2; (ii) If U is a neighborhood of p and q 2 U, there exists a neighborhood V of q so that V ⊂ U. A topology is Hausdorff if any distinct points p 6= q admit disjoint neigh- borhoods. This is almost always assumed. A set C ⊂ X is closed if its complement is open. The closure A¯ of a set A ⊂ X is the intersection of all closed sets containing X. A subset A ⊂ X is dense in X if A¯ = X. A point x 2 X is a cluster point of a subset A ⊂ X if any neighborhood of x contains a point of A distinct from x. If A0 denotes the set of cluster points, then A¯ = A [ A0: A map f : X ! Y of topological spaces is continuous at p 2 X if for any open neighborhood V ⊂ Y of f(p), there exists an open neighborhood U ⊂ X of p so that f(U) ⊂ V . -
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. -
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. -
SOME REMARKS on Sn-METRIZABLE SPACES1
Novi Sad J. Math. Vol. 39, No. 2, 2009, 103-109 SOME REMARKS ON sn-METRIZABLE SPACES1 Xun Ge2, Jinjin Li3 Abstract. This paper shows that sn-metrizable spaces can not be characterized as sequence-covering, compact, σ-images of metric spaces or sn-open, ¼, σ-images of metric spaces. Also, a space with a locally countable sn-network need not to be an sn-metrizable space. These results correct some errors on sn-metrizable spaces. AMS Mathematics Subject Classi¯cation (2000): 54C05, 54E35, 54E40 Key words and phrases: sn-metrizable space, sequence-covering mapping, sn-network, sn-open mapping, compact-mapping, ¼-mapping, σ-mapping 1. Introduction sn-metrizable spaces are a class of important generalized metric spaces be- tween g-metrizable spaces and @-spaces [9]. To characterize sn-metrizable spaces by certain images of metric spaces is an interesting question, and many \nice" characterizations of sn-metrizable spaces have been obtained [3, 8, 9, 10, 11, 14, 15, 21, 24]. In the past years, the following results were given. Proposition 1. Let X be a space. (1) X is an sn-metrizable space i® X is a sequence-covering, compact, σ- image of a metric space [21, Theorem 3.2]. (2) X is an sn-metrizable space i® X is an sn-open, ¼, σ-image of a metric space [14, Theorem 2.7]. (3) If X has a locally countable sn-network, then X is an sn-metrizable space [21, Theorem 4.3]. Unfortunately, Proposition 1.1 is not true. In this paper, we give some ex- amples to show that sn-metrizable spaces can not be characterized as sequence- covering, compact, σ-images of metric spaces or sn-open, ¼, σ-images of metric spaces, and a space with a locally countable sn-network need not to be an sn-metrizable space. -
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. -
TOPOLOGICAL SPACES DEFINITIONS and BASIC RESULTS 1.Topological Space
PART I: TOPOLOGICAL SPACES DEFINITIONS AND BASIC RESULTS 1.Topological space/ open sets, closed sets, interior and closure/ basis of a topology, subbasis/ 1st ad 2nd countable spaces/ Limits of sequences, Hausdorff property. Prop: (i) condition on a family of subsets so it defines the basis of a topology on X. (ii) Condition on a family of open subsets to define a basis of a given topology. 2. Continuous map between spaces (equivalent definitions)/ finer topolo- gies and continuity/homeomorphisms/metric and metrizable spaces/ equiv- alent metrics vs. quasi-isometry. 3. Topological subspace/ relatively open (or closed) subsets. Product topology: finitely many factors, arbitrarily many factors. Reference: Munkres, Ch. 2 sect. 12 to 21 (skip 14) and Ch. 4, sect. 30. PROBLEMS PART (A) 1. Define a non-Hausdorff topology on R (other than the trivial topol- ogy) 1.5 Is the ‘finite complement topology' on R Hausdorff? Is this topology finer or coarser than the usual one? What does lim xn = a mean in this topology? (Hint: if b 6= a, there is no constant subsequence equal to b.) 2. Let Y ⊂ X have the induced topology. C ⊂ Y is closed in Y iff C = A \ Y , for some A ⊂ X closed in X. 2.5 Let E ⊂ Y ⊂ X, where X is a topological space and Y has the Y induced topology. Thenn E (the closure of E in the induced topology on Y ) equals E \ Y , the intersection of the closure of E in X with the subset Y . 3. Let E ⊂ X, E0 be the set of cluster points of E. -
Metric Spaces
Chapter 1. Metric Spaces Definitions. A metric on a set M is a function d : M M R × → such that for all x, y, z M, Metric Spaces ∈ d(x, y) 0; and d(x, y)=0 if and only if x = y (d is positive) MA222 • ≥ d(x, y)=d(y, x) (d is symmetric) • d(x, z) d(x, y)+d(y, z) (d satisfies the triangle inequality) • ≤ David Preiss The pair (M, d) is called a metric space. [email protected] If there is no danger of confusion we speak about the metric space M and, if necessary, denote the distance by, for example, dM . The open ball centred at a M with radius r is the set Warwick University, Spring 2008/2009 ∈ B(a, r)= x M : d(x, a) < r { ∈ } the closed ball centred at a M with radius r is ∈ x M : d(x, a) r . { ∈ ≤ } A subset S of a metric space M is bounded if there are a M and ∈ r (0, ) so that S B(a, r). ∈ ∞ ⊂ MA222 – 2008/2009 – page 1.1 Normed linear spaces Examples Definition. A norm on a linear (vector) space V (over real or Example (Euclidean n spaces). Rn (or Cn) with the norm complex numbers) is a function : V R such that for all · → n n , x y V , x = x 2 so with metric d(x, y)= x y 2 ∈ | i | | i − i | x 0; and x = 0 if and only if x = 0(positive) i=1 i=1 • ≥ cx = c x for every c R (or c C)(homogeneous) • | | ∈ ∈ n n x + y x + y (satisfies the triangle inequality) Example (n spaces with p norm, p 1). -
Quantum General Relativity and the Classification of Smooth Manifolds
Quantum general relativity and the classification of smooth manifolds Hendryk Pfeiffer∗ DAMTP, Wilberforce Road, Cambridge CB3 0WA, United Kingdom Emmanuel College, St Andrew’s Street, Cambridge CB2 3AP, United Kingdom May 17, 2004 Abstract The gauge symmetry of classical general relativity under space-time diffeomorphisms implies that any path integral quantization which can be interpreted as a sum over space- time geometries, gives rise to a formal invariant of smooth manifolds. This is an oppor- tunity to review results on the classification of smooth, piecewise-linear and topological manifolds. It turns out that differential topology distinguishes the space-time dimension d = 3 + 1 from any other lower or higher dimension and relates the sought-after path integral quantization of general relativity in d = 3 + 1 with an open problem in topology, namely to construct non-trivial invariants of smooth manifolds using their piecewise-linear structure. In any dimension d ≤ 5 + 1, the classification results provide us with trian- gulations of space-time which are not merely approximations nor introduce any physical cut-off, but which rather capture the full information about smooth manifolds up to diffeomorphism. Conditions on refinements of these triangulations reveal what replaces block-spin renormalization group transformations in theories with dynamical geometry. The classification results finally suggest that it is space-time dimension rather than ab- arXiv:gr-qc/0404088v2 17 May 2004 sence of gravitons that renders pure gravity in d = 2 + 1 a ‘topological’ theory. PACS: 04.60.-m, 04.60.Pp, 11.30.-j keywords: General covariance, diffeomorphism, quantum gravity, spin foam model 1 Introduction Space-time diffeomorphisms form a gauge symmetry of classical general relativity.