ON UNIFORM TOPOLOGY and ITS APPLICATIONS 1. Introduction The
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ON METRIC SPACES AND UNIFORMITIES A THESIS SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE BY JOHN H. HARRIS DEPARTMENT OF MATHEMATICS ATLANTA, GEORGIA Ml 1966 RrU \ ~ 35 ACKNCMLEDGI*ENTS The most lucid presentation in English, to my knowledge, of the concept of a uniform space is found in John Kelly's classic, "General Topology". This text was trost influential in my choice and arrangement of materials for this thesis. I would also like to acknowledge and thank my instructors for their assistance and suggestions. A special thanks is extended to my parents without whose inspiration and encouragement this thesis would not have been possible. ii CONTENTS Page ACKNOWLEDGMENTS ii INTRODUCTION I Chapter I. Uniform Spaces « 3 1.1 Uniformities and The Uniform Topology 1.2 Uniform Continuity 1.3 The Metrization Theorem 1*JU Compactness II. Uniform Spaces and Topological Groups ... 23 III, Selected Problems ••..,.•.••...2? APPENDIX , . 31 INDEX OF SPECIAL SYMBOLS 3h BIBLIOGRAPHY 36 INTRODUCTION One of the major advantages of the theory of topology is that it lends itself readily to intuitive interpretation. This is especially true of metric spaces because of their close connection with the real number system and because of our intuitive concept of distance associated with that system. However, if we should attempt to generalize metric spaces, and we have, our intuition would often fail us for we would also have to generalize our already intuitive concept of distance. In this thesis, I shall attempt to present in a lucid and in an intuition appealing fashion, the generalized theory of metric spaces,“i.e., the theory of uniform spaces. -
Arxiv:0810.1295V3 [Math.GR] 13 Jan 2011 Hc Sijciebtntsretv.Wt Hstriooya Hand, Sa at by Terminology Rephrased Set Be This a May with That Sets M Infinite Surjective
EXPANSIVE ACTIONS ON UNIFORM SPACES AND SURJUNCTIVE MAPS TULLIO CECCHERINI-SILBERSTEIN AND MICHEL COORNAERT Abstract. We present a uniform version of a result of M. Gromov on the surjunctivity of maps commuting with expansive group ac- tions and discuss several applications. We prove in particular that for any group Γ and any field K, the space of Γ-marked groups G such that the group algebra K[G] is stably finite is compact. 1. Introduction A map f from a set X into itself is said to be surjunctive if it is surjective or not injective [Gro]. Thus, a non-surjunctive map is a map which is injective but not surjective. With this terminology at hand, Dedekind’s characterization of infinite sets may be rephrased by saying that a set X is infinite if and only if it admits a non-surjunctive map f : X → X. Similarly, elementary linear algebra tells us that a vector space is infinite-dimensional if and only if it admits a non-surjunctive endomorphism. In fact, it turns out that the absence of non-surjunctive endomorphisms for a given mathematical object X often reflects some “finiteness” property of X. The word surjunctive was created by W. Gottschalk [Got] who intro- duced the notion of a surjunctive group. Let G be a group. Given a set A, consider the set AG equipped with the prodiscrete topology, that is, with the product topology obtained by taking the discrete topology on each factor A of AG. There is a natural action of G on AG obtained by shifting coordinates via left multiplication in G (see Section 5). -
Dynamics of Infinite-Dimensional Groups and Ramsey-Type Phenomena
\rtp-impa" i i 2005/5/12 page 1 i i Dynamics of infinite-dimensional groups and Ramsey-type phenomena Vladimir Pestov May 12, 2005 i i i i \rtp-impa" i i 2005/5/12 page 2 i i i i i i \rtp-impa" i i 2005/5/12 page i i i Contents Preface iii 0 Introduction 1 1 The Ramsey{Dvoretzky{Milman phenomenon 17 1.1 Finite oscillation stability . 17 1.2 First example: the sphere S1 . 27 1.2.1 Finite oscillation stability of S1 . 27 1.2.2 Concentration of measure on spheres . 32 1.3 Second example: finite Ramsey theorem . 41 1.4 Counter-example: ordered pairs . 47 2 The fixed point on compacta property 49 2.1 Extremely amenable groups . 49 2.2 Three main examples . 55 2.2.1 Example: the unitary group . 55 2.2.2 Example: the group Aut (Q; ) . 60 ≤ 2.2.3 Counter-example: the infinite symmetric group 62 2.3 Equivariant compactification . 63 2.4 Essential sets and the concentration property . 66 2.5 Veech theorem . 71 2.6 Big sets . 76 2.7 Invariant means . 81 3 L´evy groups 95 3.1 Unitary group with the strong topology . 95 i i i i i \rtp-impa" i i 2005/5/12 page ii i i ii CONTENTS 3.2 Group of measurable maps . 98 3.2.1 Construction and example . 98 3.2.2 Martingales . 101 3.2.3 Unitary groups of operator algebras . 110 3.3 Groups of transformations of measure spaces . 115 3.3.1 Maurey's theorem . -
On Bitopological Spaces Marcus John Saegrove Iowa State University
CORE Metadata, citation and similar papers at core.ac.uk Provided by Digital Repository @ Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1971 On bitopological spaces Marcus John Saegrove Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons Recommended Citation Saegrove, Marcus John, "On bitopological spaces " (1971). Retrospective Theses and Dissertations. 4914. https://lib.dr.iastate.edu/rtd/4914 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. 71-26,888 SAEGROVE, Marcus John, 1944- ON BITOPOLOGICAL SPACES. Iowa State University, Ph.D., 1971 Mathematics University Microfilms, A XEROX Company, Ann Arbor, Michigan THIS DISSERTATION HAS BEOi MICROFILMED EXACTLY AS RECEIVED On bitopological spaces by Marcus John Saegrove A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: Mathematics Approved : Signature was redacted for privacy. In Charge oàMajor Work Signature was redacted for privacy. Signature was redacted for privacy. Iowa State University Of Science and Technology Ames, Iowa 1971 PLEASE NOTE: Some pages have indistinct print. Filmed as received. UNIVERSITY MICROFILMS. ii TABLE OF CONTENTS Page I. INTRODUCTION 1 II. SEPARATION PROPERTIES 4 III. AN INTERNAL CHARACTERIZATION OF PAIRWISE COMPLETE REGULARITY l8 IV. PRODUCTS AND BICOMPACTIFICATIONS 26 V. -
Extremely Amenable Groups and Banach Representations
Extremely Amenable Groups and Banach Representations A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Javier Ronquillo Rivera May 2018 © 2018 Javier Ronquillo Rivera. All Rights Reserved. 2 This dissertation titled Extremely Amenable Groups and Banach Representations by JAVIER RONQUILLO RIVERA has been approved for the Department of Mathematics and the College of Arts and Sciences by Vladimir Uspenskiy Professor of Mathematics Robert Frank Dean, College of Arts and Sciences 3 Abstract RONQUILLO RIVERA, JAVIER, Ph.D., May 2018, Mathematics Extremely Amenable Groups and Banach Representations (125 pp.) Director of Dissertation: Vladimir Uspenskiy A long-standing open problem in the theory of topological groups is as follows: [Glasner-Pestov problem] Let X be compact and Homeo(X) be endowed with the compact-open topology. If G ⊂ Homeo(X) is an abelian group, such that X has no G-fixed points, does G admit a non-trivial continuous character? In this dissertation we discuss some reformulations of this problem and its connections to other mathematical objects such as extremely amenable groups. When G is the closure of the group generated by a single map T ∈ Homeo(X) (with respect to the compact-open topology) and the action of G on X is minimal, the existence of non-trivial continuous characters of G is linked to the existence of equicontinuous factors of (X, T ). In this dissertation we present some connections between weakly mixing dynamical systems, continuous characters on groups, and the space of maximal chains of subcontinua of a given compact space. -
Expansive Actions with Specification on Uniform Spaces, Topological Entropy, and the Myhill Property
EXPANSIVE ACTIONS WITH SPECIFICATION ON UNIFORM SPACES, TOPOLOGICAL ENTROPY, AND THE MYHILL PROPERTY TULLIO CECCHERINI-SILBERSTEIN AND MICHEL COORNAERT Abstract. We prove that every expansive continuous action with the weak specification property of an amenable group G on a compact Hausdorff space X has the Myhill property, i.e., every pre-injective continuous self-mapping of X commuting with the action of G on X is surjective. This extends a result previously obtained by Hanfeng Li in the case when X is metrizable. Contents 1. Introduction 2 2. Background material 4 2.1. General notation 4 2.2. Subsets of X × X 4 2.3. Uniform spaces 5 2.4. Ultrauniform spaces 6 2.5. Actions 7 2.6. Covers 7 2.7. Shift spaces 8 2.8. Amenable groups 8 3. Expansiveness 9 4. Topological entropy 10 4.1. Topological entropy 10 4.2. (F, U)-separated subsets, (F, U)-spanning subsets, and (F, U)-covers 12 4.3. Uniformapproachestotopologicalentropy 14 4.4. Topological entropy and expansiveness 17 5. Homoclinicity 19 6. Weak specification 20 arXiv:1905.02740v3 [math.DS] 6 Mar 2020 7. Proof of the main result 25 References 31 Date: March 9, 2020. 2010 Mathematics Subject Classification. 54E15, 37B05, 37B10, 37B15, 37B40, 43A07, 68Q80. Key words and phrases. uniform space, uniformly continuous action, expansiveness, specification, amenable group, entropy, homoclinicity, strongly irreducible subshift, cellular automaton. 1 2 TULLIO CECCHERINI-SILBERSTEIN AND MICHEL COORNAERT 1. Introduction A topological dynamical system is a pair (X,G), where X is a topological space and G is a group acting continuously on X. -
Non-Embeddability of the Urysohn Space Into Superreflexive Banach Spaces
NON-EMBEDDABILITY OF THE URYSOHN SPACE INTO SUPERREFLEXIVE BANACH SPACES ADRIANE KAÏCHOUH Abstract. We present Pestov’s proof that the Urysohn space does not embed uniformly into a superreflexive Banach space ([P]). Its interest lies mainly in the fact that the argument is essentially combinatorial. Pestov uses the extension property for the class of finite metric spaces ([S2]) to build affine representations of the isometry group of the Urysohn space. Contents 1. Uniform embeddings 1 2. The Urysohn space 2 2.1. Katětov spaces 2 2.2. Tower construction of the Urysohn space 3 3. The extension property 4 4. Ultrapowers of Banach spaces 5 4.1. Ultrafilters 5 4.2. Ultraproducts of metric spaces 6 5. Superreflexive Banach spaces 7 6. Averaging distances 8 7. Obstruction to a uniform embedding 10 8. Concluding remarks 10 8.1. Coarse embeddability 11 8.2. Isometric embeddability 11 References 11 1. Uniform embeddings We recall the notion of uniform embedding of metric spaces. Let X and Y be two metric spaces. A uniform embedding of X into Y is an embedding of X into Y as uniform spaces. Equivalently, a map f : X ! Y is a uniform embedding if there exist two non-decreasing functions ρ1 and ρ2 from R+ to R+, with 0 < ρ1 6 ρ2 and limr!0 ρ2(r) = 0, such that for all x, x0 in X, one has 0 0 0 ρ1(dX (x; x )) 6 dY (f(x); f(x )) 6 ρ2(dX (x; x )): In particular, a uniform embedding is uniformly continuous. Uniform embeddings transpose the local structure of metric spaces: what matter are small neighborhoods of points. -
Completeness and Related Topics in a Quasi-Uniform Space
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Missouri University of Science and Technology (Missouri S&T): Scholars' Mine Scholars' Mine Doctoral Dissertations Student Theses and Dissertations 1970 Completeness and related topics in a quasi-uniform space John Warnock Carlson Follow this and additional works at: https://scholarsmine.mst.edu/doctoral_dissertations Part of the Mathematics Commons Department: Mathematics and Statistics Recommended Citation Carlson, John Warnock, "Completeness and related topics in a quasi-uniform space" (1970). Doctoral Dissertations. 2144. https://scholarsmine.mst.edu/doctoral_dissertations/2144 This thesis is brought to you by Scholars' Mine, a service of the Missouri S&T Library and Learning Resources. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. COMPLETENESS AND RELATED TOPICS IN A QUASI-UNIFORM SPACE by JOHN WARNOCK CARLSON, 1940 - A DISSERTATION Presented to the Faculty of the Graduate School of the UNIVERSITY OF MISSOURI - ROLLA In Partial Fulfillment of the Requirements for t h e Degree DOCTOR OF PHILOSOPHY in MATHEMATICS 1970 ii ABSTRACT Completions and a strong completion of a quasi- uniform space are constructed and examined. It is shown that the trivial completion of a To space is T 0 • Ex amples are given to show that a T 1 space need not have a T 1 strong completion and a T 2 space need not have a T 2 completion. The nontrivial completion constructed is shown to be T 1 if the space is T 1 and the quasi-uniform structure is the Pervin structure. -
Quasi-Actions on Trees I. Bounded Valence
Annals of Mathematics, 158 (2003), 115–164 Quasi-actions on trees I. Bounded valence By Lee Mosher, Michah Sageev, and Kevin Whyte Abstract Given a bounded valence, bushy tree T , we prove that any cobounded quasi-action of a group G on T is quasiconjugate to an action of G on an- other bounded valence, bushy tree T ′. This theorem has many applications: quasi-isometric rigidity for fundamental groups of finite, bushy graphs of coarse PD(n) groups for each fixed n; a generalization to actions on Cantor sets of Sullivan’s theorem about uniformly quasiconformal actions on the 2-sphere; and a characterization of locally compact topological groups which contain a virtually free group as a cocompact lattice. Finally, we give the first exam- ples of two finitely generated groups which are quasi-isometric and yet which cannot act on the same proper geodesic metric space, properly discontinuously and cocompactly by isometries. 1. Introduction A quasi-action of a group G on a metric space X associates to each g ∈ G a quasi-isometry Ag: x → g · x of X, with uniform quasi-isometry constants, so that AId = IdX , and so that the distance between Ag ◦ Ah and Agh in the sup norm is uniformly bounded independent of g, h ∈ G. Quasi-actions arise naturally in geometric group theory: if a metric space X is quasi-isometric to a finitely generated group G with its word metric, then the left action of G on itself can be “quasiconjugated” to give a quasi- action of G on X. Moreover, a quasi-action which arises in this manner is arXiv:math/0010136v2 [math.GR] 23 Feb 2004 cobounded and proper; these properties are generalizations of cocompact and properly discontinuous as applied to isometric actions. -
A Totally Bounded Uniformity on Coarse Metric Spaces
A totally bounded Uniformity on coarse metric Spaces Elisa Hartmann July 8, 2019 Abstract This paper presents a new version of boundary on coarse spaces. The space of ends func- tor maps coarse metric spaces to uniform topological spaces and coarse maps to uniformly continuous maps. Contents 0 Introduction 1 0.1 BackgroundandrelatedTheories . ....... 2 0.2 MainContributions............................... .... 3 0.3 Outline ......................................... 4 1 Metric Spaces 4 2 Totally Bounded Uniformity 6 3 Definition 10 4 Properties 13 5 Side Notes 18 6 Remarks 20 0 Introduction Coarse Geometry of metric spaces studies the large scale properties of a metric space. Meanwhile uniformity of metric spaces is about small scale properties. Our purpose is to pursue a new version of duality between the coarse geometry of metric spaces and uniform spaces. We present a notion of boundary on coarse metric spaces which is a arXiv:1712.02243v4 [math.MG] 5 Jul 2019 totally bounded separating uniform space. The methods are very basic and do not require any deep theory. Note that the topology of metric spaces is well understood and there are a number of topo- logical tools that can be applied on coarse metric spaces which have not been used before. The new discovery may lead to new insight on the topic of coarse geometry. 1 0 INTRODUCTION Elisa Hartmann 0.1 Background and related Theories There are quite number of notions for a boundary on a metric space. In this chapter we are going to discuss properties for three of them. The first paragraph is denoted to the Higson corona, in the second paragraph the space of ends is presented and in the last paragraph we study the Gromov boundary. -
Metrization of Uniform Spaces
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by K-State Research Exchange METRIZATION OF UNIFORM SPACES by SAMUEL LELAND LESSSIG S., Fort Hays Kansas State College, 19&1 A MASTER'S REPORT submitted in partial fulfillment of the requirements for the degree MASTER OF SCIENCE Department of Mathematics KANSAS STATS UNIVERSITY Manhattan, Kansas 1963 Approved by: ft* Docw r-v, */WS TABLE 0? CONTENTS INTRODUCTION 1 RELATIONS 2 DEFINTION OF UNIFORM SPACE 3 DEFINTION OF BASE AMD SUBBASE k DEFINTION OF UNIFORM TOPOLOGY 6 UNIFORM CONTINUITY 9 PRODUCT UNIFORMITIES 12 METRIZATION 15 ACKNOWLEDGEMENT 21 REFERENCES 22 INTRODUCTION thesis in 1906 I 4 Frechet first considered abstract spaces in his J. Much of the developement of the concept of a topological space may be found in 1 6~] Kausdorff s C-rundzur-:e ler Mcngeniehre f and in the early volumes of Fundamenta Kathematica . From this research two fundamental concepts have developed: that of a toplogical space and that of a uniform space. Andre Weil was the first to formalize the notion of a uniform space in a paper in 193? [WJ* The concept of a uniform space developed from the study of topological groups and has presented many new ideas to the study of metric spaces. One of these is the concept that the finest uniformity for a metrizable space is the one in- . using uniform spaces in duced by a metric ^12 J J. R. Isbell has done work ~| the study of the algebra of functions f? • This report is devoted to the examination of several properties of uni- form spaces and the relationship between uniformities and pseudo-metrics. -
Localic Completion of Uniform Spaces∗ 1
Logical Methods in Computer Science Vol. 13(3:22)2017, pp. 1–39 Submitted Mar. 14, 2016 www.lmcs-online.org Published Sep. 13, 2017 LOCALIC COMPLETION OF UNIFORM SPACES ∗ TATSUJI KAWAI School of Information Science, Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, Ishikawa 923-1292, Japan e-mail address: [email protected] Abstract. We extend the notion of localic completion of generalised metric spaces by Steven Vickers to the setting of generalised uniform spaces. A generalised uniform space (gus) is a set X equipped with a family of generalised metrics on X, where a generalised metric on X is a map from X × X to the upper reals satisfying zero self-distance law and triangle inequality. For a symmetric generalised uniform space, the localic completion lifts its generalised uniform structure to a point-free generalised uniform structure. This point-free structure induces a complete generalised uniform structure on the set of formal points of the localic completion that gives the standard completion of the original gus with Cauchy filters. We extend the localic completion to a full and faithful functor from the category of locally compact uniform spaces into that of overt locally compact completely regular formal topologies. Moreover, we give an elementary characterisation of the cover of the localic completion of a locally compact uniform space that simplifies the existing characterisation for metric spaces. These results generalise the corresponding results for metric spaces by Erik Palmgren. Furthermore, we show that the localic completion of a symmetric gus is equivalent to the point-free completion of the uniform formal topology associated with the gus.