Lecture 2: Basics Topics in Computational Topology: an Algorithmic View
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The Topology of Subsets of Rn
Lecture 1 The topology of subsets of Rn The basic material of this lecture should be familiar to you from Advanced Calculus courses, but we shall revise it in detail to ensure that you are comfortable with its main notions (the notions of open set and continuous map) and know how to work with them. 1.1. Continuous maps “Topology is the mathematics of continuity” Let R be the set of real numbers. A function f : R −→ R is called continuous at the point x0 ∈ R if for any ε> 0 there exists a δ> 0 such that the inequality |f(x0) − f(x)| < ε holds for all x ∈ R whenever |x0 − x| <δ. The function f is called contin- uous if it is continuous at all points x ∈ R. This is basic one-variable calculus. n Let R be n-dimensional space. By Or(p) denote the open ball of radius r> 0 and center p ∈ Rn, i.e., the set n Or(p) := {q ∈ R : d(p, q) < r}, where d is the distance in Rn. A function f : Rn −→ R is called contin- n uous at the point p0 ∈ R if for any ε> 0 there exists a δ> 0 such that f(p) ∈ Oε(f(p0)) for all p ∈ Oδ(p0). The function f is called continuous if it is continuous at all points p ∈ Rn. This is (more advanced) calculus in several variables. A set G ⊂ Rn is called open in Rn if for any point g ∈ G there exists a n δ> 0 such that Oδ(g) ⊂ G. -
TOPOLOGY and ITS APPLICATIONS the Number of Complements in The
TOPOLOGY AND ITS APPLICATIONS ELSEVIER Topology and its Applications 55 (1994) 101-125 The number of complements in the lattice of topologies on a fixed set Stephen Watson Department of Mathematics, York Uniuersity, 4700 Keele Street, North York, Ont., Canada M3J IP3 (Received 3 May 1989) (Revised 14 November 1989 and 2 June 1992) Abstract In 1936, Birkhoff ordered the family of all topologies on a set by inclusion and obtained a lattice with 1 and 0. The study of this lattice ought to be a basic pursuit both in combinatorial set theory and in general topology. In this paper, we study the nature of complementation in this lattice. We say that topologies 7 and (T are complementary if and only if 7 A c = 0 and 7 V (T = 1. For simplicity, we call any topology other than the discrete and the indiscrete a proper topology. Hartmanis showed in 1958 that any proper topology on a finite set of size at least 3 has at least two complements. Gaifman showed in 1961 that any proper topology on a countable set has at least two complements. In 1965, Steiner showed that any topology has a complement. The question of the number of distinct complements a topology on a set must possess was first raised by Berri in 1964 who asked if every proper topology on an infinite set must have at least two complements. In 1969, Schnare showed that any proper topology on a set of infinite cardinality K has at least K distinct complements and at most 2” many distinct complements. -
Higher Dimensional Conundra
Higher Dimensional Conundra Steven G. Krantz1 Abstract: In recent years, especially in the subject of harmonic analysis, there has been interest in geometric phenomena of RN as N → +∞. In the present paper we examine several spe- cific geometric phenomena in Euclidean space and calculate the asymptotics as the dimension gets large. 0 Introduction Typically when we do geometry we concentrate on a specific venue in a particular space. Often the context is Euclidean space, and often the work is done in R2 or R3. But in modern work there are many aspects of analysis that are linked to concrete aspects of geometry. And there is often interest in rendering the ideas in Hilbert space or some other infinite dimensional setting. Thus we want to see how the finite-dimensional result in RN changes as N → +∞. In the present paper we study some particular aspects of the geometry of RN and their asymptotic behavior as N →∞. We choose these particular examples because the results are surprising or especially interesting. We may hope that they will lead to further studies. It is a pleasure to thank Richard W. Cottle for a careful reading of an early draft of this paper and for useful comments. 1 Volume in RN Let us begin by calculating the volume of the unit ball in RN and the surface area of its bounding unit sphere. We let ΩN denote the former and ωN−1 denote the latter. In addition, we let Γ(x) be the celebrated Gamma function of L. Euler. It is a helpful intuition (which is literally true when x is an 1We are happy to thank the American Institute of Mathematics for its hospitality and support during this work. -
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.......................... -
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). -
Ball, Deborah Loewenberg Developing Mathematics
DOCUMENT RESUME ED 399 262 SP 036 932 AUTHOR Ball, Deborah Loewenberg TITLE Developing Mathematics Reform: What Don't We Know about Teacher Learning--But Would Make Good Working Hypotheses? NCRTL Craft Paper 95-4. INSTITUTION National Center for Research on Teacher Learning, East Lansing, MI. SPONS AGENCY Office of Educational Research and Improvement (ED), Washington, DC. PUB DATE Oct 95 NOTE 53p.; Paper presented at a conference on Teacher Enhancement in Mathematics K-6 (Arlington, VA, November 18-20, 1994). AVAILABLE FROMNational Center for Research on Teacher Learning, 116 Erickson Hall, Michigan State University, East Lansing, MI 48824-1034. PUB TYPE Reports Research/Technical (143) Speeches /Conference Papers (150) EDRS PRICE MF01/PC03 Plus Postage. DESCRIPTORS *Educational Change; *Educational Policy; Elementary School Mathematics; Elementary Secondary Education; *Faculty Development; *Knowledge Base for Teaching; *Mathematics Instruction; Mathematics Teachers; Secondary School Mathematics; *Teacher Improvement ABSTRACT This paper examines what teacher educators, policymakers, and teachers think they know about the current mathematics reforms and what it takes to help teachers engage with these reforms. The analysis is organized around three issues:(1) the "it" envisioned by the reforms;(2) what teachers (and others) bring to learning "it";(3) what is known and believed about teacher learning. The second part of the paper deals with what is not known about the reforms and how to help a larger number of teachers engage productively with the reforms. This section of the paper confronts the current pressures to "scale up" reform efforts to "reach" more teachers. Arguing that what is known is not sufficient to meet the demand for scaling up, three potential hypotheses about teacher learning and professional development are proposed that might serve to meet the demand to offer more teachers opportunities for learning. -
Simplices in the Euclidean Ball Matthieu Fradelizi, Grigoris Paouris, Carsten Schütt
Simplices in the Euclidean ball Matthieu Fradelizi, Grigoris Paouris, Carsten Schütt To cite this version: Matthieu Fradelizi, Grigoris Paouris, Carsten Schütt. Simplices in the Euclidean ball. Canadian Mathematical Bulletin, 2011, 55 (3), pp.498-508. 10.4153/CMB-2011-142-1. hal-00731269 HAL Id: hal-00731269 https://hal-upec-upem.archives-ouvertes.fr/hal-00731269 Submitted on 12 Sep 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Simplices in the Euclidean ball Matthieu Fradelizi, Grigoris Paouris,∗ Carsten Sch¨utt To appear in Canad. Math. Bull. Abstract We establish some inequalities for the second moment 1 2 |x|2dx |K| ZK of a convex body K under various assumptions on the position of K. 1 Introduction The starting point of this paper is the article [2], where it was shown that if all the extreme points of a convex body K in Rn have Euclidean norm greater than r > 0, then 1 r2 x 2dx > (1) K | |2 9n | | ZK where x 2 stands for the Euclidean norm of x and K for the volume of K. | | | | r2 We improve here this inequality showing that the optimal constant is , n + 2 with equality for the regular simplex, with vertices on the Euclidean sphere of radius r. -
01. Review of Metric Spaces and Point-Set Topology 1. Euclidean
(September 28, 2018) 01. Review of metric spaces and point-set topology Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http:=/www.math.umn.edu/egarrett/m/real/notes 2018-19/01 metric spaces and topology.pdf] 1. Euclidean spaces 2. Metric spaces 3. Completions of metric spaces 4. Topologies of metric spaces 5. General topological spaces 6. Compactness and sequential compactness 7. Total-boundedness criterion for compact closure 8. Baire's theorem 9. Appendix: mapping-property characterization of completions 1. Euclidean spaces n Let R be the usual Euclidean n-space, that is, ordered n-tuples x = (x1; : : : ; xn) of real numbers. In addition to vector addition (termwise) and scalar multiplication, we have the usual distance function on Rn, in coordinates x = (x1; : : : ; xn) and y = (y1; : : : ; yn), defined by p 2 2 d(x; y) = (x1 − y1) + ::: + (xn − yn) Of course there is visible symmetry d(x; y) = d(y; x), and positivity: d(x; y) = 0 only for x = y. The triangle inequality d(x; z) ≤ d(x; y) + d(y; z) is not trivial to prove. In the one-dimensional case, the triangle inequality is an inequality on absolute values, and can be proven case-by-case. In Rn, it is best to use the following set-up. The usual inner product (or dot-product) on Rn is x · y = hx; yi = h(x1; : : : ; xn); (y1; : : : ; yn)i = x1y1 + ::: + xnyn and jxj2 = hx; xi. Context distinguishes the norm jxj of x 2 Rn from the usual absolute value jcj on real or complex numbers c. -
N-Dimensional Quasipolar Coordinates - Theory and Application
UNLV Theses, Dissertations, Professional Papers, and Capstones 5-1-2014 N-Dimensional Quasipolar Coordinates - Theory and Application Tan Mai Nguyen University of Nevada, Las Vegas Follow this and additional works at: https://digitalscholarship.unlv.edu/thesesdissertations Part of the Mathematics Commons Repository Citation Nguyen, Tan Mai, "N-Dimensional Quasipolar Coordinates - Theory and Application" (2014). UNLV Theses, Dissertations, Professional Papers, and Capstones. 2125. http://dx.doi.org/10.34917/5836144 This Thesis is protected by copyright and/or related rights. It has been brought to you by Digital Scholarship@UNLV with permission from the rights-holder(s). You are free to use this Thesis in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Thesis has been accepted for inclusion in UNLV Theses, Dissertations, Professional Papers, and Capstones by an authorized administrator of Digital Scholarship@UNLV. For more information, please contact [email protected]. N-DIMENSIONAL QUASIPOLAR COORDINATES - THEORY AND APPLICATION By Tan Nguyen Bachelor of Science in Mathematics University of Nevada Las Vegas 2011 A thesis submitted in partial fulfillment of the requirements for the Master of Science - Mathematical Sciences Department of Mathematical Sciences College of Sciences The Graduate -
Solutions to Problem Set 4: Connectedness
Solutions to Problem Set 4: Connectedness Problem 1 (8). Let X be a set, and T0 and T1 topologies on X. If T0 ⊂ T1, we say that T1 is finer than T0 (and that T0 is coarser than T1). a. Let Y be a set with topologies T0 and T1, and suppose idY :(Y; T1) ! (Y; T0) is continuous. What is the relationship between T0 and T1? Justify your claim. b. Let Y be a set with topologies T0 and T1 and suppose that T0 ⊂ T1. What does connectedness in one topology imply about connectedness in the other? c. Let Y be a set with topologies T0 and T1 and suppose that T0 ⊂ T1. What does one topology being Hausdorff imply about the other? d. Let Y be a set with topologies T0 and T1 and suppose that T0 ⊂ T1. What does convergence of a sequence in one topology imply about convergence in the other? Solution 1. a. By definition, idY is continuous if and only if preimages of open subsets of (Y; T0) are open subsets of (Y; T1). But, the preimage of U ⊂ Y is U itself. Hence, the map is continuous if and only if open subsets of (Y; T0) are open in (Y; T1), i.e. T0 ⊂ T1, i.e. T1 is finer than T0. ` b. If Y is disconnected in T0, it can be written as Y = U V , where U; V 2 T0 are non-empty. Then U and V are also open in T1; hence, Y is disconnected in T1 too. Thus, connectedness in T1 imply connectedness in T0. -
Natural Topology
Natural Topology Frank Waaldijk ú —with great support from Wim Couwenberg ‡ ú www.fwaaldijk.nl/mathematics.html ‡ http://members.chello.nl/ w.couwenberg ∼ Preface to the second edition In the second edition, we have rectified some omissions and minor errors from the first edition. Notably the composition of natural morphisms has now been properly detailed, as well as the definition of (in)finite-product spaces. The bibliography has been updated (but remains quite incomplete). We changed the names ‘path morphism’ and ‘path space’ to ‘trail morphism’ and ‘trail space’, because the term ‘path space’ already has a well-used meaning in general topology. Also, we have strengthened the part of applied mathematics (the APPLIED perspective). We give more detailed representations of complete metric spaces, and show that natural morphisms are efficient and ubiquitous. We link the theory of star-finite metric developments to efficient computing with morphisms. We hope that this second edition thus provides a unified frame- work for a smooth transition from theoretical (constructive) topology to ap- plied mathematics. For better readability we have changed the typography. The Computer Mod- ern fonts have been replaced by the Arev Sans fonts. This was no small operation (since most of the symbol-with-sub/superscript configurations had to be redesigned) but worthwhile, we believe. It would be nice if more fonts become available for LATEX, the choice at this moment is still very limited. (the author, 14 October 2012) Copyright © Frank Arjan Waaldijk, 2011, 2012 Published by the Brouwer Society, Nijmegen, the Netherlands All rights reserved First edition, July 2011 Second edition, October 2012 Cover drawing ocho infinito xxiii by the author (‘ocho infinito’ in co-design with Wim Couwenberg) Summary We develop a simple framework called ‘natural topology’, which can serve as a theoretical and applicable basis for dealing with real-world phenom- ena. -
Knowing Mathematics for Teaching
Knowing Mathematics for Teaching Who Knows Mathematics Well Enough To Teach Third Grade, and How Can We Decide? By Deborah Loewenberg Ball, Heather C. Hill, and Hyman Bass ith the release of every new international mathe- implement standards and curriculum effectively, school sys- matics assessment, concern about U.S. students’ tems depend upon the work of skilled teachers who under- Wmathematics achievement has grown. Each stand the subject matter. How well teachers know mathe- mediocre showing by American students makes it plain that matics is central to their capacity to use instructional mate- the teaching and learning of mathematics needs improve- rials wisely, to assess students’ progress, and to make sound ment. Thus, the country, once more, has begun to turn its judgments about presentation, emphasis, and sequencing. worried attention to mathematics education. Unfortunately, That the quality of mathematics teaching depends on past reform movements have consisted more of effort than teachers’ knowledge of the content should not be a surprise. effect. We are not likely to succeed this time, either, with- Equally unsurprising is that many U.S. teachers lack sound out accounting for the disappointing outcomes of past ef- mathematical understanding and skill. This is to be ex- forts and examining the factors that contribute to success in pected because most teachers—like most other adults in other countries. Consider what research and experience this country—are graduates of the very system that we seek consistently reveal: Although the typical methods of im- to improve. Their own opportunities to learn mathematics proving U.S. instructional quality have been to develop cur- have been uneven, and often inadequate, just like those of riculum, and—especially in the last decade—to articulate their non-teaching peers.