Reconstruction of Compacta by Finite Approximations and Inverse
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Metric Geometry in a Tame Setting
University of California Los Angeles Metric Geometry in a Tame Setting A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Erik Walsberg 2015 c Copyright by Erik Walsberg 2015 Abstract of the Dissertation Metric Geometry in a Tame Setting by Erik Walsberg Doctor of Philosophy in Mathematics University of California, Los Angeles, 2015 Professor Matthias J. Aschenbrenner, Chair We prove basic results about the topology and metric geometry of metric spaces which are definable in o-minimal expansions of ordered fields. ii The dissertation of Erik Walsberg is approved. Yiannis N. Moschovakis Chandrashekhar Khare David Kaplan Matthias J. Aschenbrenner, Committee Chair University of California, Los Angeles 2015 iii To Sam. iv Table of Contents 1 Introduction :::::::::::::::::::::::::::::::::::::: 1 2 Conventions :::::::::::::::::::::::::::::::::::::: 5 3 Metric Geometry ::::::::::::::::::::::::::::::::::: 7 3.1 Metric Spaces . 7 3.2 Maps Between Metric Spaces . 8 3.3 Covers and Packing Inequalities . 9 3.3.1 The 5r-covering Lemma . 9 3.3.2 Doubling Metrics . 10 3.4 Hausdorff Measures and Dimension . 11 3.4.1 Hausdorff Measures . 11 3.4.2 Hausdorff Dimension . 13 3.5 Topological Dimension . 15 3.6 Left-Invariant Metrics on Groups . 15 3.7 Reductions, Ultralimits and Limits of Metric Spaces . 16 3.7.1 Reductions of Λ-valued Metric Spaces . 16 3.7.2 Ultralimits . 17 3.7.3 GH-Convergence and GH-Ultralimits . 18 3.7.4 Asymptotic Cones . 19 3.7.5 Tangent Cones . 22 3.7.6 Conical Metric Spaces . 22 3.8 Normed Spaces . 23 4 T-Convexity :::::::::::::::::::::::::::::::::::::: 24 4.1 T-convex Structures . -
K-THEORY of FUNCTION RINGS Theorem 7.3. If X Is A
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 69 (1990) 33-50 33 North-Holland K-THEORY OF FUNCTION RINGS Thomas FISCHER Johannes Gutenberg-Universitit Mainz, Saarstrage 21, D-6500 Mainz, FRG Communicated by C.A. Weibel Received 15 December 1987 Revised 9 November 1989 The ring R of continuous functions on a compact topological space X with values in IR or 0Z is considered. It is shown that the algebraic K-theory of such rings with coefficients in iZ/kH, k any positive integer, agrees with the topological K-theory of the underlying space X with the same coefficient rings. The proof is based on the result that the map from R6 (R with discrete topology) to R (R with compact-open topology) induces a natural isomorphism between the homologies with coefficients in Z/kh of the classifying spaces of the respective infinite general linear groups. Some remarks on the situation with X not compact are added. 0. Introduction For a topological space X, let C(X, C) be the ring of continuous complex valued functions on X, C(X, IR) the ring of continuous real valued functions on X. KU*(X) is the topological complex K-theory of X, KO*(X) the topological real K-theory of X. The main goal of this paper is the following result: Theorem 7.3. If X is a compact space, k any positive integer, then there are natural isomorphisms: K,(C(X, C), Z/kZ) = KU-‘(X, Z’/kZ), K;(C(X, lR), Z/kZ) = KO -‘(X, UkZ) for all i 2 0. -
1.1.5 Hausdorff Spaces
1. Preliminaries Proof. Let xn n N be a sequence of points in X convergent to a point x X { } 2 2 and let U (f(x)) in Y . It is clear from Definition 1.1.32 and Definition 1.1.5 1 2F that f − (U) (x). Since xn n N converges to x,thereexistsN N s.t. 1 2F { } 2 2 xn f − (U) for all n N.Thenf(xn) U for all n N. Hence, f(xn) n N 2 ≥ 2 ≥ { } 2 converges to f(x). 1.1.5 Hausdor↵spaces Definition 1.1.40. A topological space X is said to be Hausdor↵ (or sepa- rated) if any two distinct points of X have neighbourhoods without common points; or equivalently if: (T2) two distinct points always lie in disjoint open sets. In literature, the Hausdor↵space are often called T2-spaces and the axiom (T2) is said to be the separation axiom. Proposition 1.1.41. In a Hausdor↵space the intersection of all closed neigh- bourhoods of a point contains the point alone. Hence, the singletons are closed. Proof. Let us fix a point x X,whereX is a Hausdor↵space. Denote 2 by C the intersection of all closed neighbourhoods of x. Suppose that there exists y C with y = x. By definition of Hausdor↵space, there exist a 2 6 neighbourhood U(x) of x and a neighbourhood V (y) of y s.t. U(x) V (y)= . \ ; Therefore, y/U(x) because otherwise any neighbourhood of y (in particular 2 V (y)) should have non-empty intersection with U(x). -
The Non-Hausdorff Number of a Topological Space
THE NON-HAUSDORFF NUMBER OF A TOPOLOGICAL SPACE IVAN S. GOTCHEV Abstract. We call a non-empty subset A of a topological space X finitely non-Hausdorff if for every non-empty finite subset F of A and every family fUx : x 2 F g of open neighborhoods Ux of x 2 F , \fUx : x 2 F g 6= ; and we define the non-Hausdorff number nh(X) of X as follows: nh(X) := 1 + supfjAj : A is a finitely non-Hausdorff subset of Xg. Using this new cardinal function we show that the following three inequalities are true for every topological space X d(X) (i) jXj ≤ 22 · nh(X); 2d(X) (ii) w(X) ≤ 2(2 ·nh(X)); (iii) jXj ≤ d(X)χ(X) · nh(X) and (iv) jXj ≤ nh(X)χ(X)L(X) is true for every T1-topological space X, where d(X) is the density, w(X) is the weight, χ(X) is the character and L(X) is the Lindel¨ofdegree of the space X. The first three inequalities extend to the class of all topological spaces d(X) Pospiˇsil'sinequalities that for every Hausdorff space X, jXj ≤ 22 , 2d(X) w(X) ≤ 22 and jXj ≤ d(X)χ(X). The fourth inequality generalizes 0 to the class of all T1-spaces Arhangel ski˘ı’s inequality that for every Hausdorff space X, jXj ≤ 2χ(X)L(X). It is still an open question if 0 Arhangel ski˘ı’sinequality is true for all T1-spaces. It follows from (iv) that the answer of this question is in the afirmative for all T1-spaces with nh(X) not greater than the cardianilty of the continuum. -
MTH 304: General Topology Semester 2, 2017-2018
MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds.......................... -
Compact Non-Hausdorff Manifolds
A.G.M. Hommelberg Compact non-Hausdorff Manifolds Bachelor Thesis Supervisor: dr. D. Holmes Date Bachelor Exam: June 5th, 2014 Mathematisch Instituut, Universiteit Leiden 1 Contents 1 Introduction 3 2 The First Question: \Can every compact manifold be obtained by glueing together a finite number of compact Hausdorff manifolds?" 4 2.1 Construction of (Z; TZ ).................................5 2.2 The space (Z; TZ ) is a Compact Manifold . .6 2.3 A Negative Answer to our Question . .8 3 The Second Question: \If (R; TR) is a compact manifold, is there always a surjective ´etale map from a compact Hausdorff manifold to R?" 10 3.1 Construction of (R; TR)................................. 10 3.2 Etale´ maps and Covering Spaces . 12 3.3 A Negative Answer to our Question . 14 4 The Fundamental Groups of (R; TR) and (Z; TZ ) 16 4.1 Seifert - Van Kampen's Theorem for Fundamental Groupoids . 16 4.2 The Fundamental Group of (R; TR)........................... 19 4.3 The Fundamental Group of (Z; TZ )........................... 23 2 Chapter 1 Introduction One of the simplest examples of a compact manifold (definitions 2.0.1 and 2.0.2) that is not Hausdorff (definition 2.0.3), is the space X obtained by glueing two circles together in all but one point. Since the circle is a compact Hausdorff manifold, it is easy to prove that X is indeed a compact manifold. It is also easy to show X is not Hausdorff. Many other examples of compact manifolds that are not Hausdorff can also be created by glueing together compact Hausdorff manifolds. -
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). -
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 . -
INTRODUCTION to ALGEBRAIC TOPOLOGY 1 Category And
INTRODUCTION TO ALGEBRAIC TOPOLOGY (UPDATED June 2, 2020) SI LI AND YU QIU CONTENTS 1 Category and Functor 2 Fundamental Groupoid 3 Covering and fibration 4 Classification of covering 5 Limit and colimit 6 Seifert-van Kampen Theorem 7 A Convenient category of spaces 8 Group object and Loop space 9 Fiber homotopy and homotopy fiber 10 Exact Puppe sequence 11 Cofibration 12 CW complex 13 Whitehead Theorem and CW Approximation 14 Eilenberg-MacLane Space 15 Singular Homology 16 Exact homology sequence 17 Barycentric Subdivision and Excision 18 Cellular homology 19 Cohomology and Universal Coefficient Theorem 20 Hurewicz Theorem 21 Spectral sequence 22 Eilenberg-Zilber Theorem and Kunneth¨ formula 23 Cup and Cap product 24 Poincare´ duality 25 Lefschetz Fixed Point Theorem 1 1 CATEGORY AND FUNCTOR 1 CATEGORY AND FUNCTOR Category In category theory, we will encounter many presentations in terms of diagrams. Roughly speaking, a diagram is a collection of ‘objects’ denoted by A, B, C, X, Y, ··· , and ‘arrows‘ between them denoted by f , g, ··· , as in the examples f f1 A / B X / Y g g1 f2 h g2 C Z / W We will always have an operation ◦ to compose arrows. The diagram is called commutative if all the composite paths between two objects ultimately compose to give the same arrow. For the above examples, they are commutative if h = g ◦ f f2 ◦ f1 = g2 ◦ g1. Definition 1.1. A category C consists of 1◦. A class of objects: Obj(C) (a category is called small if its objects form a set). We will write both A 2 Obj(C) and A 2 C for an object A in C. -
Classification of Compact 2-Manifolds
Virginia Commonwealth University VCU Scholars Compass Theses and Dissertations Graduate School 2016 Classification of Compact 2-manifolds George H. Winslow Virginia Commonwealth University Follow this and additional works at: https://scholarscompass.vcu.edu/etd Part of the Geometry and Topology Commons © The Author Downloaded from https://scholarscompass.vcu.edu/etd/4291 This Thesis is brought to you for free and open access by the Graduate School at VCU Scholars Compass. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of VCU Scholars Compass. For more information, please contact [email protected]. Abstract Classification of Compact 2-manifolds George Winslow It is said that a topologist is a mathematician who can not tell the difference between a doughnut and a coffee cup. The surfaces of the two objects, viewed as topological spaces, are homeomorphic to each other, which is to say that they are topologically equivalent. In this thesis, we acknowledge some of the most well-known examples of surfaces: the sphere, the torus, and the projective plane. We then ob- serve that all surfaces are, in fact, homeomorphic to either the sphere, the torus, a connected sum of tori, a projective plane, or a connected sum of projective planes. Finally, we delve into algebraic topology to determine that the aforementioned sur- faces are not homeomorphic to one another, and thus we can place each surface into exactly one of these equivalence classes. Thesis Director: Dr. Marco Aldi Classification of Compact 2-manifolds by George Winslow Bachelor of Science University of Mary Washington Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in the Department of Mathematics Virginia Commonwealth University 2016 Dedication For Lily ii Abstract It is said that a topologist is a mathematician who can not tell the difference between a doughnut and a coffee cup. -
Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces Leonard Duane Wilkins [email protected]
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Tennessee, Knoxville: Trace University of Tennessee, Knoxville Trace: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 5-2011 Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces Leonard Duane Wilkins [email protected] Recommended Citation Wilkins, Leonard Duane, "Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces. " PhD diss., University of Tennessee, 2011. https://trace.tennessee.edu/utk_graddiss/1039 This Dissertation is brought to you for free and open access by the Graduate School at Trace: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of Trace: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Leonard Duane Wilkins entitled "Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Mathematics. Conrad P. Plaut, Major Professor We have read this dissertation and recommend its acceptance: James Conant, Fernando Schwartz, Michael Guidry Accepted for the Council: Dixie L. Thompson Vice Provost and Dean of the Graduate School (Original signatures are on file with official student records.) To the Graduate Council: I am submitting herewith a dissertation written by Leonard Duane Wilkins entitled \Discrete Geometric Homotopy Theory and Critical Values of Metric Spaces." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Mathematics. -
A Structure Theorem in Finite Topology
Canad. Math. Bull. Vol. 26 (1), 1983 A STRUCTURE THEOREM IN FINITE TOPOLOGY BY JOHN GINSBURG ABSTRACT. Simple structure theorems or representation theorems make their appearance at a very elementary level in subjects such as algebra, but one infrequently encounters such theorems in elemen tary topology. In this note we offer one such theorem, for finite topological spaces, which involves only the most elementary con cepts: discrete space, indiscrete space, product space and homeomorphism. Our theorem describes the structure of those finite spaces which are homogeneous. In this short note we will establish a simple representation theorem for finite topological spaces which are homogeneous. We recall that a topological space X is said to be homogeneous if for all points x and y of X there is a homeomorphism from X onto itself which maps x to y. For any natural number k we let D(k) and I(k) denote the set {1, 2,..., k} with the discrete topology and the indiscrete topology respectively. As usual, if X and Y are topological spaces then XxY is regarded as a topological space with the product topology, which has as a basis all sets of the form GxH where G is open in X and H is open in Y. If the spaces X and Y are homeomorphic we write X^Y. The number of elements of a set S is denoted by \S\. THEOREM. Let X be a finite topological space. Then X is homogeneous if and only if there exist integers m and n such that X — D(m)x/(n).