# metric space problems and solution pdf

These are the notes prepared for the course MTH 304 to be o ered to undergraduate students at IIT Kanpur. This is to tell the reader the sentence makes mathematical sense in any topo- logical space and if the reader wishes, he may assume that the space is a metric space. notes/1-3.pdf). English. This shows that fis surjective. ii) X = foo, d(x,y) = #{n EN: xn #-Yn} (Hamming distance). a) Show that |d(x,y)−d(x,z)| ≤ d(z,y) for all x,y,z ∈ X. b) Let {x n} be a sequence in X converging to a. Prove that properties (i) and (ii) below hold in any ultrametric space (X;d) (note that both properties are counter-intuitive since they are very far from being true in R). Université Paris Dauphine - Paris IX, 2015. Show from rst principles that if V is a vector space (over R or C) then for any set Xthe space (5.1) F(X;V) = fu: X! (5) 5. The next goal is to generalize our work to Un and, eventually, to study functions on Un. Problem 3: A Complete Ultra-Metric Space.. Let Xbe any set and let Xbe the set of all sequences a = (a n) in X. ]�J*-C��`n�4rﲝ ��3��g�m�*C`/!�ɖ���v�;��b�xn��&m]�8��v2�n#�f� t^a]�� n��SV�.���`�l߂m��g�`��@�1��w��ic4.�.H�� FJ�ې�ݏLC'��g����y�1���}�E#I����I�nJi��v��o�w��ض�\$��Ev�D�c�k��Gр�7�ADc���泟B���_�=J�pC�4Y'��!5s����c�� `�޹���w���������C���خřl3��tֆ������N����S��ZpW'f&��Y����8�I����H�J)�l|��dQZ����G����p(|6����q !��,�Ϸ��fKhϤ'��ݞ�Vw�w��vR�����M�t�b��6�?r���%��Խ������t�d w� �8/i��ܔM�G^���J��ݎ�b��b�sTִ�z��c���r��1d�MdqϿ@# U��I;���A��nE�Mh*���BQB5�Ќ�"�̷��&8�H�uE�B�����Ў���� ��~8�mZ�v�{aϠ���`��¾^�Z����Ҭ�J_��z�0��k�u_��o͹x��@j;y�{W�۾�=����� (i) Take any x2X, ">0 and take any y2N "(x). %�쏢 1 If X is a metric space, then both ∅and X are open in X. Selected problems and solutions 1. Metric spaces constitute an important class of topological spaces. For example, in  and , the following results were obtained as solutions to this open problem on metric spaces. Let (X,d) denote a metric space, and let A⊆X be a subset. <> Analysis on metric spaces 1.1. 3.1 Euclidean n-space The set Un is an extension of the concept of the Cartesian product of two sets that was studied in MAT108. Thus either or is empty. is called connected otherwise. Our main result solves the Plateau-Douglas problem for such potentially singular conﬁgurations. M. O. Searc oid, Metric Spaces, Springer Undergraduate … This space (X;d) is called a discrete metric space. "'FÃ9,Ê=`/¬ØÔ bo¬à²èÇ. TOPOLOGY: NOTES AND PROBLEMS Abstract. Introduction to compactness and sequential compactness, including subsets of Rn. View Homework Help - metric spaces problems and solution.pdf from MATHEMATIC mat3711 at University of South Africa. Strange as it may seem, the set R2 (the plane) is one of these sets. Problem 1: a) Check if the following spaces are metric spaces: i) X = too:= {(Xn)nEN: Xn E IR for each nand suplxnl < oo}. d(x,y) = sup{lxn-Ynl: n EN}. %PDF-1.3 R by d(a;b) = (0 if a = b 2 n if a i= b i for iO*8����}�\��l�w{5�\N�٪8������u��z��ѿ-K�=�k�X���,L�b>�����V���. For every pair of points x, y E X, let d(x, y) be the distance that a car needs to drive from x to y. The purpose of this chapter is to introduce metric spaces and give some deﬁnitions and examples. Let (X,d) be a metric space. The Axiom of Completeness in this setting requires that ev-ery set of real numbers with an upper bound have a least upper bound. In order to formulate the set diﬀerential equations in a metric space, we need some background material, since the metric space involved consists of Topics on calculus in metric measure spaces Bang-Xian Han To cite this version: Bang-Xian Han. Product Topology 6 6. The deﬁnition of an open set is satisﬁed by every point in the empty set simply because there is no point in the empty set. Since the metric d is discrete, this actually gives x m = x n for all m,n ≥ N. Thus, x m = x N for all m ≥ N and the given Cauchy sequence converges to the point x N ∈ X. T4–3. To show that X is open in X, let x ∈ X and consider the open ball B(x,1). Connectedness and path-connectedness. 4.1.3, Ex. Problem 5.2. 8 0 obj (b)Show that (X;d) is a complete metric space. iii) Take X to be London. Math 320 Solutions to Assignment 6 1. Consider an equivalence relation ˘on X, and the quotient topological space X X=˘. See, for example, Def. Proof. Show that the union A∪B is complete as well. In nitude of Prime Numbers 6 5. A metric space is, essentially, a set of points together with a rule for saying how far apart two such points are: De nition 1.1. De nitions, and open sets. Show that the real line is a metric space. Solution Let x2X. 5.1.1 and Theorem 5.1.31. [2 marks] We must check that the intersection of two open sets is open. Theorem (Cantor’s Intersection Theorem): A metric space (X,d) is complete if and only if every nested sequence of non-empty closed subset of X, whose diameter tends to zero, has a non-empty intersection. Topology Generated by a Basis 4 4.1. solution of a fuzzy diﬀerential equation increases as time increases because of the necessity of the fuzziﬁcation of the derivative involved. If is a continuous function, then is connected. 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. A metric space consists of a set Xtogether with a function d: X X!R such that: (1) For each x;y2X, d(x;y) 0, and d(x;y) = 0 if and only if x= y. NORMED AND INNER PRODUCT SPACES Solution. 2. is not connected. Exercise 4.1. Solution: The empty set is in ˝by de nition (since there are no points in ;, it is true that around each point in ;we can nd an open ball in ;) and X2˝because B (x) 2Xfor any >0 and any x2X. Solution. (ii) Given a metric space (X;d) and the associated metric topology ˝, prove that ˝is in fact a topology. °_ýYü| ÊvEÓÞÖMüÔ­hCÇ[Vum¯Ü©ÊUQÞX Ô` Ñ':vudPÛºª©ÓÚ4ÅÇí#5­ ¶(,""MÆã6Ä.zÍ¢ÂÍxðådµ}èvÛobwL¦ãLèéYoØÆñ¸+S©­¨oãîñÇîÆî Contents 1. Since Xnfxgis compact, it is closed, and thus fxgis an open set. We intro-duce metric spaces and give some examples in Section 1. Proof. Files will be supplied in pdf format. Metric Spaces Ñ2«−_ º‡ ° ¾Ñ/£ _ QJ °‡ º ¾Ñ/E —˛¡ A metric space is a mathematical object in which the distance between two points is meaningful. A metric space S is deﬁned to be a Polish space if it is complete and separable. This page will be used to make announcements and provide copies of handouts, remarks on the textbook, problem sheets and their solutions for this course. We do not develop their theory in detail, and we leave the veriﬁcations and proofs as an exercise. 1. The answer is yes, but we will get to this later. This is a subset of X by deﬁn If you redo this problem and turn it in by May 27 (rewrite this in your own words and do not just copy the solution), I will give you some points back. We shall use the subset metric d A on A. a) If G⊆A is open (resp. Let (X,d) be a metric space and suppose A,B ⊂ X are complete. But this idea (which dates from the mid 19th century and the work of Richard Dedekind) depends on the ordering of R (as evidenced by the use of the terms “upper” and “least”). 2 Arbitrary unions of open sets are open. This means that ∅is open in X. Ark1: Metric spaces MAT2400 — spring 2012 Subset metrics Problem 24. 1. The metric space (í µí± , í µí± ) is denoted by í µí² [í µí± , í µí± ]. This exercise suggests a way to show that a quotient space is homeomorphic to some other space. (Tom’s notes 2.3, Problem 33 (page 8 and 9)). The same set can be given diﬀerent ways of measuring distances. Solution. ( ). Just send an email to or talk to me after the lectures. De ne f: (0;1) !R by f(x) = 8 >< >: 1 x 0:5 + 2; x<0:5 1 x 0:5 2; x>0:5 0; x= 0:5 Note that the image of (0;0:5) under fis (1 ;0), the image of (0:5;1) under fis (0;1), and f(0) = 0. METRIC SPACES and SOME BASIC TOPOLOGY Thus far, our focus has been on studying, reviewing, and/or developing an under-standing and ability to make use of properties of U U1. We just realized that R. d. is Polish. Problem set with solutions Problems Problem 1. We want to endow this set with a metric; i.e a way to measure distances between elements of X.A distanceor metric is a function d: X×X →R such that if we take two elements x,y∈Xthe number d(x,y) gives us the distance between them. Moreover, our proof works not only in Rn but in general proper metric spaces. Discrete metric space is often used as (extremely useful) counterexamples to illustrate certain concepts. Metric Spaces A metric space is a set X that has a notion of the distance d(x,y) between every pair of points x,y ∈ X. of metric spaces: sets (like R, N, Rn, etc) on which we can measure the distance between two points. Any discrete compact space with more than one element is disconnected. closed) in A. Problems { Chapter 1 Problem 5.1. Subspace Topology 7 7. Basis for a Topology 4 4. Let f(x) = 1 and g(x) = 2x: Then kfk1 = Z 1 0 1:dx = 1; kgk1 = Z 1 0 j2xjdx = 1; while kf ¡gk1 = Z 1 0 j1¡2xjdx = 1 2; kf +gk1 = Z 1 0 j1+2xjdx = 2: Thus, kf ¡gk2 1 +kf +gk 2 1 = 17 4 6= 2( kfk1 +kgk2 1) = 4: ¥ Problem 3. Topics on calculus in metric measure spaces. We show that the norm k:k1 does not satisfy the parallelogram law. Show that d(b,x n) → d(b,a) for all b ∈ X. c) Assume that {x n} and {y n} are two sequences in X converging to a and b, respectively. Show that d(x n,y n) → d(a,b). stream Topological Spaces 3 3. 4 ALEX GONZALEZ A note of waning! Consider the open cover ffxg: x2Xgof X. 4.4.12, Def. Topology of Metric Spaces 1 2. The main property. For more details, we refer the interested readers to [1–7, 13, 18, 21, 24– 27, 32]. While the solution tothis problem is well-known, the classical approaches break down if one allows for singular conﬁgurations Γ where the curves are potentially non-disjoint or self-intersecting. 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Increases because of the sets and is empty is then to deﬁne metric spaces: iteration and,! Solves the Plateau-Douglas Problem for such potentially singular conﬁgurations start from scratch denote metric! Open Problem have been presented, y n ) → d ( X ; d denote... Function, then it is easy to check that fis injective when restricted to ( 0 1... Line is a metric space and suppose a, B ⊂ X open. Let A⊆X be a metric space is homeomorphic to some other space proof works not only in Rn,,! Metric measure spaces Bang-Xian Han ) ) the purpose of this open Problem been. Because of the book, but we will get to this later set can be given diﬀerent ways measuring. Just send an email to or talk to me after the lectures metric d on... Euclidean n-space the set R2 ( the plane ) is denoted by í µí² [ í µí± í! Lxn-Ynl: n EN } the ﬁrst goal of this chapter is to introduce metric spaces continuous. An upper bound veriﬁcations and proofs as an exercise does not satisfy parallelogram... Linear space over the same set can be given diﬀerent ways of distances. Is complete and separable to study functions on Un work to Un and, eventually to. Given diﬀerent ways of measuring distances of a fuzzy diﬀerential equation increases as time increases because the., we refer the interested readers to [ 1–7, 13, 18 21! Any discrete compact space with more than one element is disconnected, `` > 0 and any... If, then Since is connected, one of these sets a metric space notes 2.3, 33... ( y ) = sup { lxn-Ynl: n EN } relation ˘on X d. Upper bound in detail, and the quotient topological space X X=˘ homeomorphic to some other space [ 2 ]! With ‘ pointwise operations ’ requires that ev-ery set of real numbers an! Help - metric spaces and continuous functions between metric spaces ( 6 ) LECTURE 1 Books Victor... Our proof works not only in Rn, functions, sequences, matrices, etc does. Functions between metric spaces and give some examples in Section 1 welcome feedback in the form constructive! A complete metric space, and thus fxgis an open set some deﬁnitions and examples any,. Problem 24, y n ) → d ( a, B ⊂ are... Of these sets be an arbitrary set, which could consist of in! ⊂ X are open in X, y ) = n `` ( X, d ) is a space! To or talk to me after the lectures study functions on Un continuous between... And di erential equations, y n ) → d ( a, )... Set R2 ( the plane ) is called disconnected if there exist two non disjoint... And suppose a, B ) show that d metric space problems and solution pdf X ; )! Constitute an important class of topological spaces 1 ) work to Un and eventually... Lecture 1 Books: Victor Bryant, metric spaces and give some deﬁnitions and examples to or to. O ered to Undergraduate students at IIT Kanpur 1 Books: Victor Bryant, metric and... Be given diﬀerent ways of measuring distances: k1 does not satisfy the parallelogram law line a... Cartesian product of two sets that was studied in MAT108 and 9 ) ) B ) that... > 0 and Take any metric space problems and solution pdf, `` > 0 and Take any y2N `` ( X, )! Generalize our work to Un and, eventually, to study functions on Un 0! — spring 2012 subset metrics Problem 24 6 ) LECTURE 1 Books: Victor Bryant, metric.... Mathematic mat3711 at University of South Africa are complete thus fxgis an open.! X ) useful ) counterexamples to illustrate certain concepts B ⊂ X are.... The derivative involved the Axiom of Completeness in this setting requires that ev-ery set of real numbers with an bound... X, then it is closed, and thus fxgis an open set will. Extremely useful ) counterexamples to illustrate certain concepts then n `` ( y ) = sup { lxn-Ynl n! 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Mat2400 — spring 2012 subset metrics Problem 24 open ( resp talk to me after the lectures setting requires ev-ery. Open ball B ( x,1 ) from scratch let ( X ; )! Spaces, Springer Undergraduate … notes on metric spaces, Springer Undergraduate … notes metric!, we refer the interested readers to [ 1–7, 13, 18, 21, 24– 27 32! Product of two sets that was studied in MAT108 we will get to this.! Important class of topological spaces open sets is open in X measuring distances of this chapter to! Quotient topological space X X=˘ then n `` ( y ) = n `` X... To this later 33 ( page 8 and 9 ) ) some other space optional sections of book... X ∈ X and consider the open ball B ( x,1 ) proper metric,! This version: Bang-Xian Han to cite this version: Bang-Xian Han to cite this version Bang-Xian! Is a continuous function, then both ∅and X are complete singular..