Coarse Cohomology with Twisted Coefficients Elisa Hartmann
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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 . -
A Guide to Topology
i i “topguide” — 2010/12/8 — 17:36 — page i — #1 i i A Guide to Topology i i i i i i “topguide” — 2011/2/15 — 16:42 — page ii — #2 i i c 2009 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2009929077 Print Edition ISBN 978-0-88385-346-7 Electronic Edition ISBN 978-0-88385-917-9 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “topguide” — 2010/12/8 — 17:36 — page iii — #3 i i The Dolciani Mathematical Expositions NUMBER FORTY MAA Guides # 4 A Guide to Topology Steven G. Krantz Washington University, St. Louis ® Published and Distributed by The Mathematical Association of America i i i i i i “topguide” — 2010/12/8 — 17:36 — page iv — #4 i i DOLCIANI MATHEMATICAL EXPOSITIONS Committee on Books Paul Zorn, Chair Dolciani Mathematical Expositions Editorial Board Underwood Dudley, Editor Jeremy S. Case Rosalie A. Dance Tevian Dray Patricia B. Humphrey Virginia E. Knight Mark A. Peterson Jonathan Rogness Thomas Q. Sibley Joe Alyn Stickles i i i i i i “topguide” — 2010/12/8 — 17:36 — page v — #5 i i The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical Association of America was established through a generous gift to the Association from Mary P. Dolciani, Professor of Mathematics at Hunter College of the City Uni- versity of New York. In making the gift, Professor Dolciani, herself an exceptionally talented and successfulexpositor of mathematics, had the purpose of furthering the ideal of excellence in mathematical exposition. -
On Finite-Dimensional Uniform Spaces
Pacific Journal of Mathematics ON FINITE-DIMENSIONAL UNIFORM SPACES JOHN ROLFE ISBELL Vol. 9, No. 1 May 1959 ON FINITE-DIMENSIONAL UNIFORM SPACES J. R. ISBELL Introduction* This paper has two nearly independent parts, con- cerned respectively with extension of mappings and with dimension in uniform spaces. It is already known that the basic extension theorems of point set topology are valid in part, and only in part, for uniformly continuous functions. The principal contribution added here is an affirmative result to the effect that every finite-dimensional simplicial complex is a uniform ANR, or ANRU. The complex is supposed to carry the uniformity in which a mapping into it is uniformly continuous if and only if its barycentric coordinates are equiuniformly continuous. (This is a metric uniformity.) The conclusion (ANRU) means that when- ever this space μA is embedded in another uniform space μX there exist a uniform neighborhood U of A (an ε-neighborhood with respect to some uniformly continuous pseudometric) and a uniformly continuous retrac- tion r : μU -> μA, It is known that the real line is not an ARU. (Definition obvious.) Our principal negative contribution here is the proof that no uniform space homeomorphic with the line is an ARU. This is also an indication of the power of the methods, another indication being provided by the failure to settle the corresponding question for the plane. An ARU has to be uniformly contractible, but it does not have to be uniformly locally an ANRU. (The counter-example is compact metric and is due to Borsuk [2]). -
Strong Shape of Uniform Spaces
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 49 (1993) 237-249 237 North-Holland Strong shape of uniform spaces Jack Segal Department of Mathematics, University of Washington, Cl38 Padeljord Hall GN-50, Seattle, WA 98195, USA Stanislaw Spiei” Institute of Mathematics, PAN, Warsaw, Poland Bernd Giinther”” Fachbereich Mathematik, Johann Wolfgang Goethe-Universitiit, Robert-Mayer-Strafie 6-10, W-6000 Frankfurt, Germany Received 19 August 1991 Revised 16 March 1992 Abstract Segal, J., S. Spiei and B. Giinther, Strong shape of uniform spaces, Topology and its Applications 49 (1993) 237-249. A strong shape category for finitistic uniform spaces is constructed and it is shown, that certain nice properties known from strong shape theory of compact Hausdorff spaces carry over to this setting. These properties include a characterization of the new category as localization of the homotopy category, the product property and obstruction theory. Keywords: Uniform spaces, finitistic uniform spaces, strong shape, Cartesian products, localiz- ation, Samuel compactification, obstruction theory. AMS (MOS) Subj. Class.: 54C56, 54835, 55N05, 55P55, 55S35. Introduction In strong shape theory of topological spaces one encounters several instances, where the desired extension of theorems from compact spaces to more general ones either leads to difficult unsolved problems or is outright impossible. Examples are: Correspondence to: Professor J. Segal, Department of Mathematics, University of Washington, Seattle, WA 98195, USA. * This paper was written while this author was visiting the University of Washington. ** This paper was written while this author was visiting the University of Washington, and this visit was supported by a DFG fellowship. -
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). -
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. -
Notes on Metric Spaces
Notes on Metric Spaces These notes introduce the concept of a metric space, which will be an essential notion throughout this course and in others that follow. Some of this material is contained in optional sections of the book, but I will assume none of that and start from scratch. Still, you should check the corresponding sections in the book for a possibly different point of view on a few things. The main idea to have in mind is that a metric space is some kind of generalization of R in the sense that it is some kind of \space" which has a notion of \distance". Having such a \distance" function will allow us to phrase many concepts from real analysis|such as the notions of convergence and continuity|in a more general setting, which (somewhat) surprisingly makes many things actually easier to understand. Metric Spaces Definition 1. A metric on a set X is a function d : X × X ! R such that • d(x; y) ≥ 0 for all x; y 2 X; moreover, d(x; y) = 0 if and only if x = y, • d(x; y) = d(y; x) for all x; y 2 X, and • d(x; y) ≤ d(x; z) + d(z; y) for all x; y; z 2 X. A metric space is a set X together with a metric d on it, and we will use the notation (X; d) for a metric space. Often, if the metric d is clear from context, we will simply denote the metric space (X; d) by X itself. -
Magnitude Homology
Magnitude homology Tom Leinster Edinburgh A theme of this conference so far When introducing a piece of category theory during a talk: 1. apologize; 2. blame John Baez. A theme of this conference so far When introducing a piece of category theory during a talk: 1. apologize; 2. blame John Baez. 1. apologize; 2. blame John Baez. A theme of this conference so far When introducing a piece of category theory during a talk: Plan 1. The idea of magnitude 2. The magnitude of a metric space 3. The idea of magnitude homology 4. The magnitude homology of a metric space 1. The idea of magnitude Size For many types of mathematical object, there is a canonical notion of size. • Sets have cardinality. It satisfies jX [ Y j = jX j + jY j − jX \ Y j jX × Y j = jX j × jY j : n • Subsets of R have volume. It satisfies vol(X [ Y ) = vol(X ) + vol(Y ) − vol(X \ Y ) vol(X × Y ) = vol(X ) × vol(Y ): • Topological spaces have Euler characteristic. It satisfies χ(X [ Y ) = χ(X ) + χ(Y ) − χ(X \ Y ) (under hypotheses) χ(X × Y ) = χ(X ) × χ(Y ): Challenge Find a general definition of `size', including these and other examples. One answer The magnitude of an enriched category. Enriched categories A monoidal category is a category V equipped with a product operation. A category X enriched in V is like an ordinary category, but each HomX(X ; Y ) is now an object of V (instead of a set). linear categories metric spaces categories enriched posets categories monoidal • Vect • Set • ([0; 1]; ≥) categories • (0 ! 1) The magnitude of an enriched category There is a general definition of the magnitude jXj of an enriched category. -
Math 118: Topology in Metric Spaces
Math 118: Topology in Metric Spaces You may recall having seen definitions of open and closed sets, as well as other topological concepts such as limit and isolated points and density, for sets of real numbers (or even sets of points in Euclidean space). A lot of these concepts help us define in analysis what we mean by points being “close” to one another so that we can discuss convergence and continuity. We are going to re-examine all of these concepts for subsets of more general spaces: One on which we have a nicely defined distance. Definition A metric space (X, d) is a set X, whose elements we call points, along with a function d : X × X → R that satisfies (a) (b) (c) For any two points p, q ∈ X, d(p, q) is called the distance from p to q. Notes: We clearly want non-negative distances, as well as symmetry. If we think of points in the Euclidean space R2, or the complex plane, property (c) is the usual triangle inequality. Also note that for any Y ⊆ X, (Y, d) is a metric space. Examples You may have noticed that in a lot of these examples, the distance is of a special form. Consider a normed vector space (a vector space, as you learned in linear algebra, allows for the addition of the elements, as well as multiplication by a scalar. A norm is a real-valued function on the vector space that satisfies the triangle inequality, is 0 only for the 0 vector, and multiplying a vector by a scalar α just increases its norm by the factor |α|.) The norm always allows us to define a distance: for two elements v, w in the space, d(v, w) = kv −wk. -
Chapter 2 Metric Spaces and Topology
2.1. METRIC SPACES 29 Definition 2.1.29. The function f is called uniformly continuous if it is continu- ous and, for all > 0, the δ > 0 can be chosen independently of x0. In precise mathematical notation, one has ( > 0)( δ > 0)( x X) ∀ ∃ ∀ 0 ∈ ( x x0 X d (x , x0) < δ ), d (f(x ), f(x)) < . ∀ ∈ { ∈ | X 0 } Y 0 Definition 2.1.30. A function f : X Y is called Lipschitz continuous on A X → ⊆ if there is a constant L R such that dY (f(x), f(y)) LdX (x, y) for all x, y A. ∈ ≤ ∈ Let fA denote the restriction of f to A X defined by fA : A Y with ⊆ → f (x) = f(x) for all x A. It is easy to verify that, if f is Lipschitz continuous on A ∈ A, then fA is uniformly continuous. Problem 2.1.31. Let (X, d) be a metric space and define f : X R by f(x) = → d(x, x ) for some fixed x X. Show that f is Lipschitz continuous with L = 1. 0 0 ∈ 2.1.3 Completeness Suppose (X, d) is a metric space. From Definition 2.1.8, we know that a sequence x , x ,... of points in X converges to x X if, for every δ > 0, there exists an 1 2 ∈ integer N such that d(x , x) < δ for all i N. i ≥ 1 n = 2 n = 4 0.8 n = 8 ) 0.6 t ( n f 0.4 0.2 0 1 0.5 0 0.5 1 − − t Figure 2.1: The sequence of continuous functions in Example 2.1.32 satisfies the Cauchy criterion. -
Metric Foundations of Geometry. I
METRIC FOUNDATIONS OF GEOMETRY. I BY GARRETT BIRKHOFF 1. Introduction. It is shown below by elementary methods that «-dimen- sional Euclidean, spherical, and hyperbolic geometry can be characterized by the following postulates. Postulate I. Space is metric (in the sense of Fréchet). Postulate II. Through any two points a line segment can be drawn, locally uniquely. Postulate III. Any isometry between subsets of space can be extended to a self-isometry of all space. The outline of the proof has been presented elsewhere (G. Birkhoff [2 ]('))• Earlier, H. Busemann [l] had shown that if, further, exactly one straight line connected any two points, and if bounded sets were compact, then the space was Euclidean or hyperbolic; moreover for this, Postulate III need only be assumed for triples of points. Later, Busemann extended the result to elliptic space, assuming that exactly one geodesic connected any two points, that bounded sets were compact, and that Postulate III held locally for triples of points (H. Busemann [2, p. 186]). We recall Lie's celebrated proof that any Riemannian variety having local free mobility is locally Euclidean, spherical, or hyperbolic. Since our Postu- late II, unlike Busemann's straight line postulates, holds in any Riemannian variety in which bounded sets are compact, our result may be regarded as freeing Lie's theorem for the first time from its analytical hypotheses and its local character. What is more important, we use direct geometric arguments, where Lie used sophisticated analytical and algebraic methods(2). Our methods are also far more direct and elementary than those of Busemann, who relies on a deep theorem of Hurewicz. -
Notes on Uniform Structures Annex to H104
Notes on Uniform Structures Annex to H104 Mariusz Wodzicki December 13, 2013 1 Vocabulary 1.1 Binary Relations 1.1.1 The power set Given a set X, we denote the set of all subsets of X by P(X) and by P∗(X) — the set of all nonempty subsets. The set of subsets E ⊆ X which contain a given subset A will be denoted PA(X). If A , Æ, then PA(X) is a filter. Note that one has PÆ(X) = P(X). 1.1.2 In these notes we identify binary relations between elements of a set X and a set Y with subsets E ⊆ X×Y of their Cartesian product X×Y. To a given relation ∼ corresponds the subset: E∼ ˜ f(x, y) 2 X×Y j x ∼ yg (1) and, vice-versa, to a given subset E ⊆ X×Y corresponds the relation: x ∼E y if and only if (x, y) 2 E.(2) 1.1.3 The opposite relation We denote by Eop ˜ f(y, x) 2 Y×X j (x, y) 2 Eg (3) the opposite relation. 1 1.1.4 The correspondence E 7−! Eop (E ⊆ X×X) (4) defines an involution1 of P(X×X). It induces the corresponding involu- tion of P(P(X×X)): op E 7−! op∗(E ) ˜ fE j E 2 E g (E ⊆ P(X×X)).(5) Exercise 1 Show that op∗(E ) (a) possesses the Finite Intersection Property, if E possesses the Finite Intersec- tion Property; (b) is a filter-base, if E is a filter-base; (c) is a filter, if E is a filter.