DOCSLIB.ORG

POINT SET TOPOLOGY Definition 1 a Topological Structure On

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
POINT SET TOPOLOGY Definition 1 a Topological Structure On POINT SET TOPOLOGY De¯nition 1 A topological structure on a set X is a family (X) called open sets and satisfying O ½ P (O ) is closed for arbitrary unions 1 O (O ) is closed for ¯nite intersections. 2 O De¯nition 2 A set with a topological structure is a topological space (X; ) O ; = 2;Ei = x : x Eifor some i = [ [i f 2 2 ;g ; so is always open by (O ) ; 1 ; = 2;Ei = x : x Eifor all i = X \ \i f 2 2 ;g so X is always open by (O2). Examples (i) = (X) the discrete topology. O P (ii) ; X the indiscrete of trivial topology. Of; g These coincide when X has one point. (iii) =the rational line. Q =set of unions of open rational intervals O De¯nition 3 Topological spaces X and X 0 are homomorphic if there is an isomorphism of their topological structures i.e. if there is a bijection (1-1 onto map) of X and X 0 which generates a bijection of and . O O e.g. If X and X are discrete spaces a bijection is a homomorphism. (see also Kelley p102 H). De¯nition 4 A base for a topological structure is a family such that B ½ O every o can be expressed as a union of sets of 2 O B Examples (i) for the discrete topological structure x x2X is a base. f g (ii) for the indiscrete topological structure ; X is a base. f; g (iii) For , topologised as before, the set of bounded open intervals is a base.Q 1 (iv) Let X = 0; 1; 2 f g Let = (0; 1); (1; 2); (0; 12) . Is this a base for some topology B f g on X? i.e. Do unions of members of X satisfy (O2)? (0; 1) (1; 2) = (1)- which is not a union of members of , so is not a\base for any topology on X. B B Theorem 1 A necessary and su±cient condition for to be a base for a 0 B 00 topology on X = o2Bo is that for each O and O and each x O0 O00 O [such that x O O0 O00. 2 B 2 \ 9 2 B 2 ½ \ Proof Necessary: If is a base for ; O0 O00 and if x O0 O00, since O0 O00 is a unionB of sets of O O \ suc2hOthat x O2 O\0 O00. \ B 9 2 B 2 ½ \ Su±cient: let be the family of unions of sets of . O B (O1) is clearly satis¯ed. (O2) ( Ai) ( Bj) = (Ai Bj) so that it is su±cient to prove that the intersection[ \ [of two [sets of\ is a union of sets of . B B Let x O0 O00. Then Ox such that x O x O0 O00 so that 0 200 \ 9 2 B 2 ½ \ O O = x2O0\O00 Ox. \ [ Theorem 2 If is a non-empty family of sets the family of their ¯nite S B intersections is a base for a topology on S [ Proof Immediate veri¯cation of O1 and O2. The topology generated in this way is the smallest topology including all the sets of . S De¯nition 4 A family is a sub base for a topology if the set of ¯nite intersections is a baseS for the topology. e.g. a a2Q and ( a) a2Q are sub bases for f 1g f ¡1 g Q De¯nition 5 If a topology has a countable base it satis¯es the second axiom of countability. De¯nition 6 In a topological space a neighbourhood of a set A is a set which contains an open set containing A. A neighbourhood of a point X is a neighbourhood of x . f g Theorem 3 A necessary and su±cient condition that a set be open is that it contains (is) a neighbourhood of each of its points. Proof Necessary: De¯nition of a neighbourhood Su±cient: Let OA = open subsets of A. OA is open (O ) and OA A. [ 1 ½ 2 If x A A a neighbourhood of x open set x therefore x OA 2 ⊃ ⊃ 3 2 therefore A OA. ½ Let V (x) denote the family of neighbourhoods of x. Then V (x) has the following properties. (V1) Every subset of X which contains a member of V (x) is a member of V (x) (V2) V (x) is closed for ¯nite intersections (V3) x belongs to every member of V (x). (V ) If v V (x) W V (x) such that v V (y) for all y W . 4 2 9 2 2 2 (Take W to be an open set x and V .) 3 ½ Theorem 4 If for each point x of X there is given a family V (x) of subsets of X, satisfying V ¡ then a unique topology on X for which the 1 4 9 sets of neighbourhoods of each point x are precisely the given V (x). (Hausdor®). Proof If such a topology exists theorem 3 shows that the open sets must be the O such that for each x O O V (x) and so there is at most one 2 2 such topology. Consider the set O so de¯ned. (O ) Suppose x O. Then x some O0 O0 V (x) O0 cupO so 1 2 [ 2 2 ½ j O V (x). [ 2 (O ) Suppose x F o x each O. each O V (x) therefore 2 2 \ ) 2 2 F O V (x) by V . \ 2 2 Now consider the system U(x) of neighbourhoods of x de¯ned by this topology. (i) U(x) V (x). Let U U(x). Then U O x. But O V (x) so by V½U V (x). 2 ⊃ 3 2 1 2 (ii) V (x) U(x). Let V V (x). It is su±cient to prove that O such that½ x O V2. 9 2 O 2 ½ Let O = y : V V (y) x O since V V (x). f 2 g 2 2 O V since y V for all y O by V . ½ 2 2 3 To prove that O it is su±cient to prove that V V (y) for all y O. 2 O 2 2 If t O V V (y) by de¯nition of O therefore W V (y) such 2 2 9 2 that V V (z) for all z W by V 4. 2 2 ¡ Therefore W O (de¯nition of O) ½ 3 Therefore O V (y) by V 2 1 e.g. R1 : V (x)= sets which contain interval (a; b) a < x < b. De¯nition 7 A metric for a set X is a function ½ form X X to R+ (non- negative reals) such that £ M1 ½(x; y) = ½(y; x) for all x; y M ½(x; z) ½(x; y) + ½(y; z) for all x; y; z 2 · M3 ½(x; x) = 0 for all x. This is sometimes called a pseudo metric and for a metric we have (M 0 ) ½(x; y) 0; = 0 x = y. 3 ¸ , De¯nition 8 The open r- Ball about x = y : ½(x; y) < r and is denoted by B(r; x) f g The closed r-ball around x = y : ½(x; y) R and is denoted by B(r; x). f · g 1 V (x) = sets which contain one of B n ; x n = 1; 2; : : : satis¯es V1¡4 and so de¯nesf a topology on X. This³ topology´ is the metricg topology de¯ned on X by ½. The development of topology from the neighbourhood point of ve=iew is due to H. Weyl and Hausdor®. That from the open sets aspect is due to Alexandro® and Hopf. De¯nition 9 The closed sets G of a topological space are the complements of the open sets. (G1) G is closed for arbitrary intersections (G ) G is closed for ¯nite unions. and X are both closed and open. 2 ; Clearly given a family G satisfying G1 and G2 the family of comple- ments is a topology for which the closed sets are the sets of G. De¯nition 10 a point x is an interior point of a set A if A is a neighbourhood of x. The set of interior points of A is the interior A0 of A. An exterior point of A is an interior point of cA (i.e. a neighbourhood of x which does not meet A, or x is isolated, separated9 from A.) Theorem 5 (i) The interior of a set is open and 4 (ii) is the largest open subset. (iii) A necessary and su±cient condition for a set to be open is that it coincides with its interior. Proof (i) x A0 open O such that x O A. 2 ) 9 2 ½ y O y A0 therefore x O A0 therefore A0 is a neigh- bourho2 o)d of2each of its points2and ½so it is open. o o 0 (ii) Let O A O A therefore O½A0 O A , but A is such an ½ ) ½ 0 [ ½ O therefore O½AO = A . [ (iii) Su±cient condition from (i). Necessary condition from (ii). 0 0 A B = A0 B0 but A B = A0 B0 z }|\ { \ z }|[ { 6 [ e.g. X = R1 with metric topology, A=rationals, B=irrationals. 0 0 0 0 A = B = therefore A B = . But A B = R1 and so 0 ; ; [ ; [ A B = R1. z }|[ { 0 However A0 B0 A B always. [ ½ z }|[ { De¯nition 11 A point x is adherent to a set A if every neighbourhood of x meets A.
Load more

Recommended publications
  • Basic Properties of Filter Convergence Spaces
    Basic Properties of Filter Convergence Spaces Barbel¨ M. R. Stadlery, Peter F. Stadlery;z;∗ yInstitut fur¨ Theoretische Chemie, Universit¨at Wien, W¨ahringerstraße 17, A-1090 Wien, Austria zThe Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA ∗Address for corresponce Abstract. This technical report summarized facts from the basic theory of filter convergence spaces and gives detailed proofs for them. Many of the results collected here are well known for various types of spaces. We have made no attempt to find the original proofs. 1. Introduction Mathematical notions such as convergence, continuity, and separation are, at textbook level, usually associated with topological spaces. It is possible, however, to introduce them in a much more abstract way, based on axioms for convergence instead of neighborhood. This approach was explored in seminal work by Choquet [4], Hausdorff [12], Katˇetov [14], Kent [16], and others. Here we give a brief introduction to this line of reasoning. While the material is well known to specialists it does not seem to be easily accessible to non-topologists. In some cases we include proofs of elementary facts for two reasons: (i) The most basic facts are quoted without proofs in research papers, and (ii) the proofs may serve as examples to see the rather abstract formalism at work. 2. Sets and Filters Let X be a set, P(X) its power set, and H ⊆ P(X). The we define H∗ = fA ⊆ Xj(X n A) 2= Hg (1) H# = fA ⊆ Xj8Q 2 H : A \ Q =6 ;g The set systems H∗ and H# are called the conjugate and the grill of H, respectively.
    [Show full text]
  • Basic Topologytaken From
    Notes by Tamal K. Dey, OSU 1 Basic Topology taken from [1] 1 Metric space topology We introduce basic notions from point set topology. These notions are prerequisites for more sophisticated topological ideas—manifolds, homeomorphism, and isotopy—introduced later to study algorithms for topological data analysis. To a layman, the word topology evokes visions of “rubber-sheet topology”: the idea that if you bend and stretch a sheet of rubber, it changes shape but always preserves the underlying structure of how it is connected to itself. Homeomorphisms offer a rigorous way to state that an operation preserves the topology of a domain, and isotopy offers a rigorous way to state that the domain can be deformed into a shape without ever colliding with itself. Topology begins with a set T of points—perhaps the points comprising the d-dimensional Euclidean space Rd, or perhaps the points on the surface of a volume such as a coffee mug. We suppose that there is a metric d(p, q) that specifies the scalar distance between every pair of points p, q ∈ T. In the Euclidean space Rd we choose the Euclidean distance. On the surface of the coffee mug, we could choose the Euclidean distance too; alternatively, we could choose the geodesic distance, namely the length of the shortest path from p to q on the mug’s surface. d Let us briefly review the Euclidean metric. We write points in R as p = (p1, p2,..., pd), d where each pi is a real-valued coordinate. The Euclidean inner product of two points p, q ∈ R is d Rd 1/2 d 2 1/2 hp, qi = Pi=1 piqi.
    [Show full text]
  • 16 Continuous Functions Definition 16.1. Let F
    Math 320, December 09, 2018 16 Continuous functions Definition 16.1. Let f : D ! R and let c 2 D. We say that f is continuous at c, if for every > 0 there exists δ>0 such that jf(x)−f(c)j<, whenever jx−cj<δ and x2D. If f is continuous at every point of a subset S ⊂D, then f is said to be continuous on S. If f is continuous on its domain D, then f is said to be continuous. Notice that in the above definition, unlike the definition of limits of functions, c is not required to be a limit point of D. It is clear from the definition that if c is an isolated point of the domain D, then f must be continuous at c. The following statement gives the interpretation of continuity in terms of neighborhoods and sequences. Theorem 16.2. Let f :D!R and let c2D. Then the following three conditions are equivalent. (a) f is continuous at c (b) If (xn) is any sequence in D such that xn !c, then limf(xn)=f(c) (c) For every neighborhood V of f(c) there exists a neighborhood U of c, such that f(U \D)⊂V . If c is a limit point of D, then the above three statements are equivalent to (d) f has a limit at c and limf(x)=f(c). x!c From the sequential interpretation, one gets the following criterion for discontinuity. Theorem 16.3. Let f :D !R and c2D. Then f is discontinuous at c iff there exists a sequence (xn)2D, such that (xn) converges to c, but the sequence f(xn) does not converge to f(c).
    [Show full text]
  • A TEXTBOOK of TOPOLOGY Lltld
    SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY lltld SEI FER T: 7'0PO 1.OG 1' 0 I.' 3- Dl M E N SI 0 N A I. FIRERED SPACES This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: SAMUELEILENBERG AND HYMANBASS A list of recent titles in this series appears at the end of this volunie. SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY H. SEIFERT and W. THRELFALL Translated by Michael A. Goldman und S E I FE R T: TOPOLOGY OF 3-DIMENSIONAL FIBERED SPACES H. SEIFERT Translated by Wolfgang Heil Edited by Joan S. Birman and Julian Eisner @ 1980 ACADEMIC PRESS A Subsidiary of Harcourr Brace Jovanovich, Publishers NEW YORK LONDON TORONTO SYDNEY SAN FRANCISCO COPYRIGHT@ 1980, BY ACADEMICPRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. 11 1 Fifth Avenue, New York. New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NWI 7DX Mit Genehmigung des Verlager B. G. Teubner, Stuttgart, veranstaltete, akin autorisierte englische Ubersetzung, der deutschen Originalausgdbe. Library of Congress Cataloging in Publication Data Seifert, Herbert, 1897- Seifert and Threlfall: A textbook of topology. Seifert: Topology of 3-dimensional fibered spaces. (Pure and applied mathematics, a series of mono- graphs and textbooks ; ) Translation of Lehrbuch der Topologic. Bibliography: p. Includes index. 1.
    [Show full text]
  • A Discussion on Analytical Study of Semi-Closed Set in Topological Space
    The International journal of analytical and experimental modal analysis ISSN NO:0886-9367 A discussion on analytical study of Semi-closed set in topological space 1 Dr. Priti Kumari, 2 Sukesh Kumar Das, 3 Dr. Ranjana & 4 Rupesh Kumar 1 & 2 Guest Assistant Professor, Department of Mathematics Saharsa College of Engineering , Saharsa ( 852201 ), Bihar, INDIA 3 University Professor, University department of Mathematics Tilka Manjhi Bhagalpur University, Bhagalpur ( 812007 ), Bihar , INDIA 4 M. Sc., Department of Physics A. N. College Patna, Univ. of Patna ( 800013 ), Bihar, INDIA [email protected] , [email protected] , [email protected] & [email protected] Abstract : In this paper, we introduce a new class of sets in the topological space, namely Semi- closed sets in the topological space. We find characterizations of these sets. Further, we study some fundamental properties of Semi-closed sets in the topological space. Keywords : Open set, Closed set, Interior of a set & Closure of a set. I. Introduction The term Semi-closed set which is a weak form of closed set in a topological space and it is introduced and defined by the mathematician N. Biswas [10] in the year 1969. The term Semi- closure of a set in a topological space defined and introduced by two mathematician Crossley S. G. & Hildebrand S. K. [3,4] in the year 1971. The mathematician N. Levine [1] also defined and studied the term generalized closed sets in the topological space in Jan 1970. The term Semi- Interior point & Semi-Limit point of a subset of a topological space was defined and studied by the mathematician P.
    [Show full text]
  • Some Properties of Θ-Open Sets
    Divulgaciones Matem¶aticasVol. 12 No. 2(2004), pp. 161{169 Some Properties of θ-open Sets Algunas Propiedades de los Conjuntos θ-abiertos M. Caldas ([email protected]) Departamento de Matematica Aplicada, Universidade Federal Fluminense, Rua Mario Santos Braga, s/n 24020-140, Niteroi, RJ Brasil. S. Jafari ([email protected]) Department of Mathematics and Physics, Roskilde University, Postbox 260, 4000 Roskilde, Denmark. M. M. Kov¶ar([email protected]) Department of Mathematics, Faculty of Electrical Engineering and Computer Sciences Technical University of Brno, Technick ¶a8 616 69 Brno, Czech Republic. Abstract In the present paper, we introduce and study topological properties of θ-derived, θ-border, θ-frontier and θ-exterior of a set using the con- cept of θ-open sets and study also other properties of the well known notions of θ-closure and θ-interior. Key words and phrases: θ-open, θ-closure, θ-interior, θ-border, θ- frontier, θ-exterior. Resumen En el presente ert¶³culose introducen y estudian las propiedades to- pol¶ogicasdel θ-derivedo, θ-borde, θ-frontera y θ-exterior de un conjunto usando el concepto de conjunto θ-abierto y estudiando tambi¶enotras propiedades de las nociones bien conocidas de θ-clausura y θ-interior. Palabras y frases clave: θ-abierto, θ-clausura, θ-interior, θ-borde, θ-frontera, θ-exterior. Received 2003/09/30. Revised 2004/10/15. Accepted 2004/10/19. MSC (2000): Primary 54A20, 54A05. 162 M. Caldas, S. Jafari, M. M. Kov¶ar 1 Introduction The notions of θ-open subsets, θ-closed subsets and θ-closure where introduced by Veli·cko [14] for the purpose of studying the important class of H-closed spaces in terms of arbitrary ¯berbases.
    [Show full text]
  • Mathematical Analysis, Second Edition
    PREFACE A glance at the table of contents will reveal that this textbooktreats topics in analysis at the "Advanced Calculus" level. The aim has beento provide a develop- ment of the subject which is honest, rigorous, up to date, and, at thesame time, not too pedantic.The book provides a transition from elementary calculusto advanced courses in real and complex function theory, and it introducesthe reader to some of the abstract thinking that pervades modern analysis. The second edition differs from the first inmany respects. Point set topology is developed in the setting of general metricspaces as well as in Euclidean n-space, and two new chapters have been addedon Lebesgue integration. The material on line integrals, vector analysis, and surface integrals has beendeleted. The order of some chapters has been rearranged, many sections have been completely rewritten, and several new exercises have been added. The development of Lebesgue integration follows the Riesz-Nagyapproach which focuses directly on functions and their integrals and doesnot depend on measure theory.The treatment here is simplified, spread out, and somewhat rearranged for presentation at the undergraduate level. The first edition has been used in mathematicscourses at a variety of levels, from first-year undergraduate to first-year graduate, bothas a text and as supple- mentary reference.The second edition preserves this flexibility.For example, Chapters 1 through 5, 12, and 13 providea course in differential calculus of func- tions of one or more variables. Chapters 6 through 11, 14, and15 provide a course in integration theory. Many other combinationsare possible; individual instructors can choose topics to suit their needs by consulting the diagram on the nextpage, which displays the logical interdependence of the chapters.
    [Show full text]
  • 3 Limits of Sequences and Filters
    3 Limits of Sequences and Filters The Axiom of Choice is obviously true, the well-ordering theorem is obviously false; and who can tell about Zorn’s Lemma? —Jerry Bona (Schechter, 1996) Introduction. Chapter 2 featured various properties of topological spaces and explored their interactions with a few categorical constructions. In this chapter we’ll again discuss some topological properties, this time with an eye toward more fine-grained ideas. As introduced early in a study of analysis, properties of nice topological spaces X can be detected by sequences of points in X. We’ll be interested in some of these properties and the extent to which sequences suffice to detect them. But take note of the adjective “nice” here. What if X is any topological space, not just a nice one? Unfortunately, sequences are not well suited for characterizing properties in arbitrary spaces. But all is not lost. A sequence can be replaced with a more general construction—a filter—which is much better suited for the task. In this chapter we introduce filters and highlight some of their strengths. Our goal is to spend a little time inside of spaces to discuss ideas that may be familiar from analysis. For this reason, this chapter contains less category theory than others. On the other hand, we’ll see in section 3.3 that filters are a bit like functors and hence like generalizations of points. This perspective thus gives us a coarse-grained approach to investigating fine-grained ideas. We’ll go through some of these basic ideas—closure, limit points, sequences, and more—rather quickly in sections 3.1 and 3.2.
    [Show full text]
  • Chapter 2 Limits of Sequences
    Chapter 2 Limits of Sequences Calculus Student: lim sn = 0 means the sn are getting closer and closer to zero but n!1 never gets there. Instructor: ARGHHHHH! Exercise 2.1 Think of a better response for the instructor. In particular, provide a counterexample: find a sequence of numbers that 'are getting closer and closer to zero' but aren't really getting close at all. What about the 'never gets there' part? Should it be necessary that sequence values are never equal to its limit? 2.1 Definition and examples We are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero Then we define what it means for sequence to converge to an arbitrary real number. Finally, we discuss the various ways a sequence may diverge (not converge). In between we will apply what we learn to further our understanding of real numbers and to develop tools that are useful for proving the important theorems of Calculus. Recall that a sequence is a function whose domain is Z+ or Z≥. A sequence is most usually denoted with subscript notation rather than standard function notation, that is we write sn rather than s(n): See Section 0.3.2 for more about definitions and notations used in describing sequences. 43 44 CHAPTER 2. LIMITS OF SEQUENCES 1 Figure 2.1: s = : n n 2 1 0 0 5 10 15 20 2.1.1 Sequences converging to zero. Definition We say that the sequence sn converges to 0 whenever the following hold: For all > 0, there exists a real number, N, such that n > N =) jsnj < .
    [Show full text]
  • Tutorial Sheet 2, Topology 2011 (With Solutions)
    Tutorial Sheet 2, Topology 2011 (with Solutions) 1. Let X be any set and p 2 X be some point in X. Define τ to be the collection of all subsets of X that do not contain p, plus X itself. Prove that τ is a topology on X. (It is called the \excluded point topology.") Solution: Just check this satisfies the definition of topology: contains the empty set, entire space, unions, and finite complements. 2 2. Consider the following metrics on R (which are not the usual metric): d1(x; y) = maxi=1;2 jxi − yij and d2(x; y) = jx1 − y1j + jx2 − y2j. Describe the open sets induced by these metrics. (What does an open ball look like?) Solution: For the first metric, the open ball of radius at the origin is a square with sides length 2, not including its edges, which are parallel to the axes, and centered at the origin. For the second metric, we also get a square but it has been rotated so that its vertices now lie on the axes, at points (±, 0) and (0; ±). One can actually check, though, that the topologies induced by these metrics are both the same as the usual topology. (Although that wasn't really part of the question.) 3. Let (X; d) be a metric space containing at least two points. Prove that the metric topology cannot be the trivial topology. Solution: We just need to find an open set other than ; or X, which implies the topology cannot be trivial. Take x; y 2 X such that x 6= y.
    [Show full text]
  • 206 APPENDIX a GLOSSARY FLOW CHARTS Category Theory & Set Theory
    206 APPENDIX A GLOSSARY FLOW CHARTS category theory & set theory injective union surjective quotient map intersection bijective inverse map set element equivalence relation image pre-image object map category 207 point-set topology compactification homeomorphism complete quotient space Hausdor↵ imbedding metric completion gluing paracompact isometry cutting interior continuous cauchy boundary subspace topology compact metric topology convergent limit point connected limit closure topology open metric sequence closed category theory & set theory 208 numbers and algebra modular arithmetic vector space torsion real numbers, R group action module Z/nZ group integers, Z ring rational numbers, Q field binary operation identity element point-set topology cardinality inverse operation metric completion natural numbers, Z+ commutative cauchy, convergence associative metric, sequence linear category theory & set theory 209 linear algebra linear isomorphisms determinants Cramer’s rule multilinear maps tensor products exterior products biinear maps symmetric skew-symmetric definite the plane, R2 linear maps 3-space, R3 bases ordered bases rank inner products n space, Rn dimension orientation − nullity invariance of domain numbers and algebra real numbers, R module, vector space ring, field 210 differential vector calculus critical values the inverse function theorem the regular value theorem the implicit function theorem the constant rank level set theorem higher order mixed partial derivatives Rm n smooth maps A Jacobian matrix in ⇥ represents the m
    [Show full text]
  • Solutions for Homework 3
    Math 424 B / 574 B Due Wednesday, Oct 21 Autumn 2015 Solutions for Homework 3 Solution for Problem 6, page 43. Let's break it up according to the different tasks. Part (1): E0 is closed. I have to show that E0 contains its limit points. Let x 2 E00 be a limit point of E0 (the double-prime notation means (E0)0, i.e. the set of limit points of the set of limit points of E; the repetition is not a typo!). Let Nr(x) be an arbitrary neighborhood of x. By the definition of a limit point, Nr(x) contains some point y 2 E0 that is different from x. Now choose s > 0 so small that Ns(y) ⊂ Nr(x) and x 62 Ns(y). For instance, you could take s < min(d(x; y); r − d(x; y)): The inequality s < d(x; y) ensures that x 62 Ns(y), while the inequality s < r − d(x; y) means 0 that for every y 2 Ns(y) we have d(x; y0) ≤ d(x; y) + d(y; y0) < d(x; y) + s < r; 0 so that y 2 Nr(x). Or in other words, Ns(y) ⊂ Nr(x), as desired. 0 Now, y itself was in E (I chose it so), which means that Ns(y) contains some point z 2 E different from y. But now we have E 3 z 2 Ns(y) ⊂ Nr(x); and z 6= x because z is in Ns(y), which doesn't contain x. In conclusion, we have found, in the arbitrary neighborhood Nr(x) of x, a point z 2 E that is different from x.
    [Show full text]