DOCSLIB.ORG

Natural Topology

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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 The Computer Mod- ern fonts have been replaced by the Arev Sans fonts.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. Natural topology is tailored to make pointwise and pointfree notions go together naturally. As a constructive theory in BISH, it gives a classical math- ematician a faithful idea of important concepts and results in intuitionism. Natural topology is well-suited for practical and computational purposes. We give several examples relevant for applied mathematics, such as the decision-support system Hawk-Eye, and various real-number representations. We compare classical mathematics (CLASS), intuitionistic mathematics (INT), recursive mathematics (RUSS), Bishop-style mathematics (BISH) and formal topology, aiming to reduce the mutual differences to their essence. To do so, our mathematical foundation must be precise and simple. There are links with physics, regarding the topological character of our physical universe. Any natural space is isomorphic to a quotient space of Baire space, which therefore is universal. We develop an elegant and concise ‘genetic induction’ scheme, and prove its equivalence on natural spaces to a formal-topological induction style. The inductive Heine-Borel property holds for ‘compact’ or ‘fanlike’ natural subspaces, including the real interval [α, β]. Inductive mor- phisms respect this Heine-Borel property, inversely. This partly solves the continuous-function problem for BISH, yet pointwise problems persist in the absence of Brouwer’s Thesis. By inductivizing the definitions, a direct correspondence with INT is obtained which allows for a translation of many intuitionistic results into BISH. We thus prove a constructive star-finitary metrization theorem which parallels the classical metrization theorem for strongly paracompact spaces. We also obtain non-metrizable Silva spaces, in infinite-dimensional topology. Natural topology gives a solid basis, we think, for further constructive study of topo- logical lattice theory, algebraic topology and infinite-dimensional topology. The final section reconsiders the question of which mathematics to choose for physics. Compactness issues also play a role here, since the question ‘can Nature produce a non-recursive sequence?’ finds a negative answer in CTphys . CTphys , if true, would seem at first glance to point to RUSS as the mathematics of choice for physics. To discuss this issue, we wax more philosophical. We also present a simple model of INT in RUSS, in the two- player game LIfE. Contents CHAPTER ZERO: Introduction 5 Introduction . .6 CHAPTER ONE: Natural Topology 9 Basic definitions and the natural reals . 10 Natural morphisms . 17 Fundamental natural spaces . 23 Applied math intermezzo: Hawk-Eye, binary and decimal reals . 28 Intuitionistic phenomena in natural topology . 31 CHAPTER TWO: Baire space is universal 34 Introduction to Baire space . 35 Lattices, trees and spreads . 37 Universal natural spaces . 40 Morphisms on spreads and spraids . 43 A direct link with intuitionism . 45 Compactness, Brouwer’s Thesis and inductive definitions . 51 CHAPTER THREE: Inductivizing our definitions 55 Induction in formal-topology style . 56 Genetic induction in Brouwer’s style . 59 Inductive Heine-Borel for (sub)fanns . 63 Inductive morphisms . 65 Pointwise problems in BISH and final inductivization . 72 (In)finite products and Tychonoff’s theorem . 77 CHAPTER FOUR: Metrizability, and natural topology in physics 83 Metric spaces and (non-)metrizability . 84 From natural to general topology . 94 Natural topology and physics . 95 APPENDIX: Additionals, Examples, Proofs 102 Closing remarks and acknowledgements . 103 Background, motivation, and recap . 106 Examples . 113 Proofs and additional definitions . 122 Constructive concepts and axioms used . 158 Additional remarks . 166 Bibliography and further reading references . 169 CHAPTER ZERO Introduction In this chapter we introduce the subject of natural topology, starting from the natural sciences. The basis for observations and measure- ments in science seems inescapably to be one of ever-increasing refinement. We do not obtain finished real numbers, but only finite approximations of ever-increasing exactitude. This type of ‘constructive’ considerations has become increasingly important with the advent of the computer. The translation of theo- retical classical mathematics to applied mathematics is often prob- lematic. Constructive mathematics incorporates finite approxima- tions in its theory, thus providing a smooth theoretical framework for applied mathematics. Also important: natural topology gives a mathematical model of the real-world topology that we encounter when measuring. This we believe to be relevant for physics. Last but not least, we believe natural topology to be relevant for the foundations of (constructive) mathematics. Introduction 6 0.0 INTRODUCTION 0.0.0 Background and motivation of this paper For a historical background and motivation of this paper, we refer the reader to the appendix, section A.1. Section A.2 of the appendix holds some nice mathematical examples which should help clarify our approach. For readability, most proofs are given in the appendix in section A.3. Constructive axioms and concepts are given and discussed in section A.4, additional remarks can be found in section A.5, and the bibliography is in section A.6. Summarizing: don’t skip the appendix! 0.0.1 Introduction to natural topology Imagine an engineer taking measure- ments of some natural physical phenomenon. With ever-increasing precision steps she arrives at ever more precise approximations of certain real num- bers. In this process she may come across two measurements which at the outset could still indicate the same real number, yet, when more precision is attained, are seen to be really apart. The interesting thing about this description lies in the hidden meaning of the word ‘real number’. Usually this meaning is taken for granted, with an intuitive image of the real line as ‘foundation’. But in mathematics —the precision language of science— the real numbers are commonly defined as equivalence classes of Cauchy-sequences of rational numbers. Then, later, a metric topology can be defined where the basic open sets are the open rational intervals. This topology is optional, as a system the real numbers are usually viewed to exist ‘on their own’. Compared to the situation that the engineer finds herself in, the above math- ematical approach is exactly the other way round. For the engineer first and only encounters a finite number of shrinking rational intervals (the measure- ments), and then regards these finite measurements as an approximation to a real number (getting better all the time if one can apply more and more precision). From a topological view, the engineer comes across the topology of rational intervals before ever seeing a real number. It turns out that many prob- lems of translating theoretical mathematics into practical applications hinge around this reversal of approach to real numbers and their natural topology. So let us go into this matter a bit more. Introduction 7 0.0.2 Classical mathematics In theoretical mathematics, theorems about the real numbers are often proved in a way which ‘disregards’ the topology. For instance, consider a theorem asserting ∀ R∃ y R[D(, y)]. Consider a reasonably algorithmic proof even which has2 the following2 form: for < 0 do A and for 0 do B. Looking at it from a topological standpoint, one could say such a proof≥ is ‘discontinuous’ in 0. When an engineer looks to imple- ment the theorem say on a computer fed by real-time data, the problem arises immediately that for real-time data the distinction < 0 or 0 cannot always be made. On data hovering around 0, the ‘method’
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Math 131: Introduction to Topology 1
    Math 131: Introduction to Topology 1 Professor Denis Auroux Fall, 2019 Contents 9/4/2019 - Introduction, Metric Spaces, Basic Notions3 9/9/2019 - Topological Spaces, Bases9 9/11/2019 - Subspaces, Products, Continuity 15 9/16/2019 - Continuity, Homeomorphisms, Limit Points 21 9/18/2019 - Sequences, Limits, Products 26 9/23/2019 - More Product Topologies, Connectedness 32 9/25/2019 - Connectedness, Path Connectedness 37 9/30/2019 - Compactness 42 10/2/2019 - Compactness, Uncountability, Metric Spaces 45 10/7/2019 - Compactness, Limit Points, Sequences 49 10/9/2019 - Compactifications and Local Compactness 53 10/16/2019 - Countability, Separability, and Normal Spaces 57 10/21/2019 - Urysohn's Lemma and the Metrization Theorem 61 1 Please email Beckham Myers at [email protected] with any corrections, questions, or comments. Any mistakes or errors are mine. 10/23/2019 - Category Theory, Paths, Homotopy 64 10/28/2019 - The Fundamental Group(oid) 70 10/30/2019 - Covering Spaces, Path Lifting 75 11/4/2019 - Fundamental Group of the Circle, Quotients and Gluing 80 11/6/2019 - The Brouwer Fixed Point Theorem 85 11/11/2019 - Antipodes and the Borsuk-Ulam Theorem 88 11/13/2019 - Deformation Retracts and Homotopy Equivalence 91 11/18/2019 - Computing the Fundamental Group 95 11/20/2019 - Equivalence of Covering Spaces and the Universal Cover 99 11/25/2019 - Universal Covering Spaces, Free Groups 104 12/2/2019 - Seifert-Van Kampen Theorem, Final Examples 109 2 9/4/2019 - Introduction, Metric Spaces, Basic Notions The instructor for this course is Professor Denis Auroux. His email is [email protected] and his office is SC539.
    [Show full text]
  • Topologies of Some New Sequence Spaces, Their Duals, and the Graphical Representations of Neighborhoods ∗ Eberhard Malkowsky A, , Vesna Velickoviˇ C´ B
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 158 (2011) 1369–1380 Contents lists available at ScienceDirect Topology and its Applications www.elsevier.com/locate/topol Topologies of some new sequence spaces, their duals, and the graphical representations of neighborhoods ∗ Eberhard Malkowsky a, , Vesna Velickoviˇ c´ b a Department of Mathematics, Faculty of Sciences, Fatih University, 34500 Büyükçekmece, Istanbul, Turkey b Department of Mathematics, Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18000 Niš, Serbia article info abstract MSC: Many interesting topologies arise in the theory of FK spaces. Here we present some new 40H05 results on the topologies of certain FK spaces and the determinations of their dual spaces. 54E35 We also apply our software package for visualization and animations in mathematics and its extensions to the representation of neighborhoods in various topologies. Keywords: © 2011 Elsevier B.V. All rights reserved. Topologies and neighborhoods FK spaces Dual spaces Visualization Computer graphics 1. Introduction This paper deals with two subjects in mathematics and computer graphics, namely the study of certain linear topological spaces and the graphical representation of neighborhoods in the considered topologies. Many important topologies arise in the theory of sequence spaces, in particular, in FK and BK spaces and their dual spaces. Here we present some new results on the topology of certain FK spaces and their dual spaces. Our results have an interesting and important application in crystallography, in particular in the growth of crystals. According to Wulff’s principle [1], the shape of a crystal is uniquely determined by its surface energy function.
    [Show full text]
  • Topology M367K
    Topology M367K Michael Starbird June 5, 2008 2 Contents 1 Cardinality and the Axiom of Choice 5 1.1 *Well-Ordering Principle . 8 1.2 *Ordinal numbers . 9 2 General Topology 11 2.1 The Real Number Line . 11 2.2 Open Sets and Topologies . 13 2.3 Limit Points and Closed Sets . 16 2.4 Interior, Exterior, and Boundary . 19 2.5 Bases . 20 2.6 *Comparing Topologies . 21 2.7 Order Topology . 22 2.8 Subspaces . 23 2.9 *Subbases . 24 3 Separation, Countability, and Covering Properties 25 3.1 Separation Properties . 25 3.2 Countability Properties . 27 3.3 Covering Properties . 29 3.4 Metric Spaces . 30 3.5 *Further Countability Properties . 31 3.6 *Further Covering Properties . 32 3.7 *Properties on the ordinals . 32 4 Maps Between Topological Spaces 33 4.1 Continuity . 33 4.2 Homeomorphisms . 35 4.3 Product Spaces . 36 4.4 Finite Products . 36 3 4 CONTENTS 4.5 *Infinite Products . 37 5 Connectedness 39 5.1 Connectedness . 40 5.2 Continua . 41 5.3 Path or Arc-Wise Connectedness . 43 5.4 Local Connectedness . 43 A The real numbers 45 B Review of Set Theory and Logic 47 B.1 Set Theory . 47 B.2 Logic . 47 Chapter 1 Cardinality and the Axiom of Choice At the end of the nineteenth century, mathematicians embarked on a pro- gram whose aim was to axiomatize all of mathematics. That is, the goal was to emulate the format of Euclidean geometry in the sense of explicitly stating a collection of definitions and unproved axioms and then proving all mathematical theorems from those definitions and axioms.
    [Show full text]
  • General Topology. Part 1: First Concepts
    A Review of General Topology. Part 1: First Concepts Wayne Aitken∗ March 2021 version This document gives the first definitions and results of general topology. It is the first document of a series which reviews the basics of general topology. Other topics such as Hausdorff spaces, metric spaces, connectedness, and compactness will be covered in further documents. General topology can be viewed as the mathematical study of continuity, con- nectedness, and related phenomena in a general setting. Such ideas first arise in particular settings such as the real line, Euclidean spaces (including infinite di- mensional spaces), and function spaces. An initial goal of general topology is to find an abstract setting that allows us to formulate results concerning continuity and related concepts (convergence, compactness, connectedness, and so forth) that arise in these more concrete settings. The basic definitions of general topology are due to Fr`echet and Hausdorff in the early 1900's. General topology is sometimes called point-set topology since it builds directly on set theory. This is in contrast to algebraic topology, which brings in ideas from abstract algebra to define algebraic invariants of spaces, and differential topology which considers topological spaces with additional structure allowing for the notion of differentiability (basic general topology only generalizes the notion of continuous functions, not the notion of dif- ferentiable function). We approach general topology here as a common foundation for many subjects in mathematics: more advanced topology, of course, including advanced point-set topology, differential topology, and algebraic topology; but also any part of analysis, differential geometry, Lie theory, and even algebraic and arith- metic geometry.
    [Show full text]
  • Differential Topology: Exercise Sheet 1
    Prof. Dr. M. Wolf WS 2018/19 M. Heinze Sheet 1 Differential Topology: Exercise Sheet 1 Exercises (for Oct. 24th and 25th) 1.1 Examples for quotient topologies Draw the following topological spaces: (a) Let V := [−1; 1] × [0; 1] ⊂ R2 with the subspace topology and ∼ the equivalence relation such that (x; 0) ∼ (x; 1) 8x 2 [−1; 1] and all other (x; t), 0 < t < 1 are only equivalent to themselves. Consider V= ∼. (b) Let W := [−1; 1] × [0; 1] ⊂ R2 with the subspace topology and ∼ the equivalence relation such that (x; 0) ∼ (−x; 1) 8x 2 [−1; 1] and all other (x; t), 0 < t < 1 are only equivalent to themselves. Consider W= ∼. (c) Define the group action (Z × Z) × R2 ! R2 as (m; n) · (x; y) 7! (x + m; y + n) for m, n 2 Z, x,y 2 R. Consider R2=(Z × Z). By this, we mean the quotient space of R2 with respect to the equivalence relation 2 (u; v) ∼ (x; y) if 9(m; n) 2 Z×Z such that (m; n)·(u; v) = (x; y) for (x; y); (u; v) 2 R : (d) Identify S1 ' fz 2 Cj jzj = 1g. Let Y := S1 × [0; 1] and ∼ the equivalence relation such that (z; 0) ∼ (z; 1) 8z 2 C and all other (z; t), 0 < t < 1 are only equivalent to themselves. Consider Y= ∼. Solution: (a) Cylinder (b) Moebius strip (c) Torus (d) Klein bottle 1.2 Hausdorff property of quotient spaces In this exercise you will show that the Hausdorff separation property of a given topological space does generally not extend to a quotient space thereof.
    [Show full text]
  • Lecture Notes in Topology
    Lecture Notes in Topology Lectures by Andrei Khrennikov Note that these are rapidly taken and then even more swiftly typed notes, and as such errors might well occur. Be sure to check any oddities against the course literature [KF20]. Last updated March 1, 2016. Throughout this document, signifies end proof, N signifies end of example, and signifies end of solution. Table of Contents Table of Contents i 1 Lecture I 1 1.1 Metric Spaces . 1 1.2 Open Sets in Metric Spaces . 1 1.3 Properties of Open Sets in Metric Spaces . 2 1.4 Topology . 3 2 Lecture II 4 2.1 More on Topology . 4 2.2 Comparison of Topologies . 5 2.3 Continuous Functions . 6 2.4 Base of Topology . 6 3 Lecture III 8 3.1 Topology from a System of Neighbourhoods . 8 3.2 Axioms of Separability . 9 3.3 Closure of a Set . 10 3.4 Compact Topological Spaces . 11 3.5 Centred Systems . 11 4 Lecture IV 12 4.1 Relative Topology . 12 5 Lecture V 13 5.1 More on Open and Closed Sets . 13 5.2 Compact Sets and Countability . 14 Notes by Jakob Streipel. i TABLE OF CONTENTS ii 6 Lecture VI 16 6.1 Compactness and Related Notions . 16 6.2 Compactness in Metric Spaces . 17 7 Lecture VII 22 7.1 Topological Concepts in Metric Spaces . 22 7.2 Relative Compactness . 24 References 25 Notations 26 Index 27 LECTURE I 1 1 Lecture I1 Note first of all that the lecture notes assumes familiarity with basic set the- oretical concepts such as taking unions and intersections of sets, differences of sets, complements, et cetera.
    [Show full text]
  • Topology Notes
    Topology Lectures –Integration workshop 2017 David Glickenstein August 2, 2017 Abstract Lecture notes from the Integration Workshop at University of Ari- zona, August 2017. These notes are based heavily on notes from previous integration workshops written by Philip Foth, Tom Kennedy, Shankar Venkataramani and others. 1 Introduction to topology 1.1 Topology theorems The basics of point set topology arise from trying to understand the following theorems from basic calculus: (in the following, we assume intervals written [a; b] have the property a < b so that they are not empty) Theorem 1 (Intermediate Value Theorem) If f :[a; b] R is a contin- uous function, and y is a number between f (a) and f (b) !then there exists x [a; b] such that f (x) = y: 2 Theorem 2 (Extreme Value Theorem) If f :[a; b] R is a continuous ! function, then there are numbers xm; xM [a; b] such that 2 f (xm) = min f (x): x [a; b] ; f 2 g f (xM ) = max f (x): x [a; b] : f 2 g Here are some other interesting theorems in topology that we will not prove here: Theorem 3 (Jordan Curve Theorem) Any continuous simple closed curve in the plane, separates the plane into two disjoint regions, the inside and the outside. Theorem 4 (Jordan-Schoen‡eiss Theorem) For any simple closed curve in the plane, there is a homeomorphism of the plane which takes that curve into the standard circle. 1 Theorem 5 (Generalized Jordan Curve Theorem) Any embedding of the (n 1)-dimensional sphere into n-dimensional Euclidean space separates the Euclidean space into two disjoint regions.
    [Show full text]
  • Introductory Notes in Topology
    Introductory notes in topology Stephen Semmes Rice University Contents 1 Topological spaces 5 1.1 Neighborhoods . 5 2 Other topologies on R 6 3 Closed sets 6 3.1 Interiors of sets . 7 4 Metric spaces 8 4.1 Other metrics . 8 5 The real numbers 9 5.1 Additional properties . 10 5.2 Diameters of bounded subsets of metric spaces . 10 6 The extended real numbers 11 7 Relatively open sets 11 7.1 Additional remarks . 12 8 Convergent sequences 12 8.1 Monotone sequences . 13 8.2 Cauchy sequences . 14 9 The local countability condition 14 9.1 Subsequences . 15 9.2 Sequentially closed sets . 15 10 Local bases 16 11 Nets 16 11.1 Sub-limits . 17 12 Uniqueness of limits 17 1 13 Regularity 18 13.1 Subspaces . 18 14 An example 19 14.1 Topologies and subspaces . 20 15 Countable sets 21 15.1 The axiom of choice . 22 15.2 Strong limit points . 22 16 Bases 22 16.1 Sub-bases . 23 16.2 Totally bounded sets . 23 17 More examples 24 18 Stronger topologies 24 18.1 Completely Hausdorff spaces . 25 19 Normality 25 19.1 Some remarks about subspaces . 26 19.2 Another separation condition . 27 20 Continuous mappings 27 20.1 Simple examples . 28 20.2 Sequentially continuous mappings . 28 21 The product topology 29 21.1 Countable products . 30 21.2 Arbitrary products . 31 22 Subsets of metric spaces 31 22.1 The Baire category theorem . 32 22.2 Sequences of open sets . 32 23 Open sets in R 33 23.1 Collections of open sets .
    [Show full text]
  • Touching the Z2 in Three-Dimensional Rotations
    VOL. 81, NO. 5, DECEMBER 2008 345 Touching the Z2 in Three-Dimensional Rotations VESNA STOJANOSKA Northwestern University Evanston IL 60201 [email protected] ORLIN STOYTCHEV American University in Bulgaria 2700 Blagoevgrad, Bulgaria and Institute for Nuclear Research 1784 Sofia, Bulgaria [email protected] Rotations, belts, braids, spin-1/2 particles, and all that The space of all three-dimensional rotations is usually denoted by SO(3). This space has a well-known and fascinating topological property—a complete rotation of an ob- ject is a motion which may or may not be continuously deformable to the trivial motion (i.e., no motion at all) but the composition of two motions that are not deformable to the trivial one gives a motion, which is. (Here and further down by “complete rota- tion” we will mean taking the object at time t = 0 and rotating it as t changes from 0 to 1 arbitrarily around a fixed point, so that at t = 1 the object is brought back to its initial orientation.) A rotation around some fixed axis by 360◦ cannot be continuously deformed to the trivial motion, but it can be deformed to a rotation by 360◦ around any other axis (in any direction). However, a rotation by 720◦ is deformable to the trivial one. You may try to see some of this at home by performing a complete rotation of a box, keeping one of the vertices fixed. Let us first rotate the box around one of the edges and then try to deform this motion to the trivial one.
    [Show full text]