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. Prove that X must be nite. If is a continuous function, then is connected. Given a metric space (X;d), a point x2Xand ">0, de ne B ... interesting example of an ultrametric space is given in the next problem. Problem 1: a) Check if the following spaces are metric spaces: i) X = too:= {(Xn)nEN: Xn E IR for each nand suplxnl < oo}. 5.1.1 and Theorem 5.1.31. Then N "(y) = N "(x). Strange as it may seem, the set R2 (the plane) is one of these sets. ��. Math 320 Solutions to Assignment 6 1. Product Topology 6 6. First, we prove 1. Our main result solves the Plateau-Douglas problem for such potentially singular conﬁgurations. We just realized that R. d. is Polish. Solution. A metric space is a set X together with a function d (called a metric or "distance function") which assigns a real number d(x, y) to every pair x, y X satisfying the properties (or axioms): d(x, y) 0 and d(x, y) = 0 x = y, d(x, y) = d(y, x), d(x, y) + d(y, z) d(x, z). tel-01178865 Universit´e Paris-Dauphine Ecole Doctorale de Dauphine´ CEREMADE TOPICS ON CALCULUS ON METRIC MEASURE SPACES … 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. Solution. 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. De nitions, and open sets. 2 Arbitrary unions of open sets are open. See, for example, Def. Solution Let x2X. 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. A metric space is called disconnected if there exist two non empty disjoint open sets : such that . The Axiom of Completeness in this setting requires that ev-ery set of real numbers with an upper bound have a least upper bound. 4.1.3, Ex. 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. TOPOLOGY: NOTES AND PROBLEMS Abstract. Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. This shows that fis surjective. I welcome feedback in the form of constructive comments or criticism. 1 If X is a metric space, then both ∅and X are open in X. M. O. Searc oid, Metric Spaces, Springer Undergraduate … 1. Since Xnfxgis compact, it is closed, and thus fxgis an open set. View Homework Help - metric spaces problems and solution.pdf from MATHEMATIC mat3711 at University of South Africa. Contents 1. A metric space S is deﬁned to be a Polish space if it is complete and separable. We do not develop their theory in detail, and we leave the veriﬁcations and proofs as an exercise. 2. is not connected. Problem 3. x��ZIo\����f �S�}�I �͛���2$��H�ѿ�W�^wW���P�
���R]����v�&�6��*q����'O�_ݸI�b����o6ߞa�����ٜ]=��7��jr�����͓��n�}v[[�`�v��FE���vn�N��M���n'�M����w)����>O*8����}�\��l�w{5�\N�٪8������u��z��ѿ-K�=�k�X���,L�b>�����V���. stream 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. Topological Spaces 3 3. Thus either or is empty. 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. Is there a countable dense subset of C[0,T ] of C[0, ∞), namely are these spaces Polish as well? d(x,y) = sup{lxn-Ynl: n EN}. Topology Generated by a Basis 4 4.1. De ne d: XX! %�쏢 General Mathematics [math.GM]. [2 marks] We must check that the intersection of two open sets is open. Problem set with solutions Problems Problem 1. 1. Proof. Some of this material is contained in optional sections of the book, but I will assume none of that and start from scratch. closed) in A. Moreover, our proof works not only in Rn but in general proper metric spaces. These are the notes prepared for the course MTH 304 to be o ered to undergraduate students at IIT Kanpur. 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. 4.4.12, Def. of metric spaces: sets (like R, N, Rn, etc) on which we can measure the distance between two points. ]�J*-C��`n�4rﲝ ��3��g�m�*C`/!�ɖ���v�;��b�xn��&m]�8��v2�n#�f�
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��~8�mZ�v�{aϠ���`��¾^�Z����Ҭ�J_��z�0��k�u_��ox��@j;y�{W�۾�=����� Solutions to Problem Sheet 4 Jos e A. Canizo~ March 2013 Unless otherwise speci ed, the symbols X, Y and Zrepresent topological spaces in the following exercises. Let (X,d) be a metric space and suppose A,B ⊂ X are complete. Basis for a Topology 4 4. If , then Since is connected, one of the sets and is empty. In nitude of Prime Numbers 6 5. We shall use the subset metric d A on A. a) If G⊆A is open (resp. Problems { Chapter 1 Problem 5.1. <> ( ). Let (X;d) be a metric space. In order to formulate the set diﬀerential equations in a metric space, we need some background material, since the metric space involved consists of 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). The metric space (í µí± , í µí± ) is denoted by í µí² [í µí± , í µí± ]. 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! 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. °_ýYü| ÊvEÓÞÖMüÔhCÇ[Vum¯Ü©ÊUQÞX Ô`
Ñ':vudPÛºª©ÓÚ4ÅÇí#5 ¶(,""MÆã6Ä.zÍ¢ÂÍxðådµ}èvÛobwL¦ãLèéYoØÆñ¸+S©¨oãîñÇîÆî NNT: 2015PA090014. Metric spaces constitute an important class of topological spaces. R by d(a;b) = (0 if a = b 2 n if a i= b i for i0 and take any y2N "(x). The next goal is to generalize our work to Un and, eventually, to study functions on Un. Connectedness and path-connectedness. 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. Closed Sets, Hausdor Spaces, … Analysis on metric spaces 1.1. Just send an email to or talk to me after the lectures. "'FÃ9,Ê=`/¬ØÔ bo¬à²èÇ. Show that d(x n,y n) → d(a,b). The contraction mapping theorem, with applications in the solution of equations and di erential equations. is called connected otherwise. 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. (6) LECTURE 1 Books: Victor Bryant, Metric spaces: iteration and application, Cambridge, 1985. The purpose of this chapter is to introduce metric spaces and give some deﬁnitions and examples. 1. NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. The answer is yes, but we will get to this later. 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. Introduction to compactness and sequential compactness, including subsets of Rn. (ii) Given a metric space (X;d) and the associated metric topology ˝, prove that ˝is in fact a topology. Some solutions of this open problem have been presented. Problem 5.2. Problems and solutions 1. Consider the open cover ffxg: x2Xgof X. A metric space is deﬁned to be separable if it contains a dense countable subset A. Problem 3: A Complete Ultra-Metric Space.. Let Xbe any set and let Xbe the set of all sequences a = (a n) in X. 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”). Selected problems and solutions 1. Topics on calculus in metric measure spaces Bang-Xian Han To cite this version: Bang-Xian Han. Topology of Metric Spaces 1 2. MAS331: Metric spaces Problems The questions that have been marked with an … English. 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. Suppose that every subset of Xis compact. Show that the union A∪B is complete as well. Proof. Université Paris Dauphine - Paris IX, 2015. We show that the norm k:k1 does not satisfy the parallelogram law. Let (X,d) be a metric space. The same set can be given diﬀerent ways of measuring distances. The ﬁrst goal of this course is then to deﬁne metric spaces and continuous functions between metric spaces. 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. Topics on calculus in metric measure spaces. This is a subset of X by deﬁn Files will be supplied in pdf format. For more details, we refer the interested readers to [1–7, 13, 18, 21, 24– 27, 32]. This space (X;d) is called a discrete metric space. 8 0 obj This exercise suggests a way to show that a quotient space is homeomorphic to some other space. Furthermore, it is easy to check that fis injective when restricted to (0;0:5) or (0:5;1). closed) in X, then it is open (resp. Ark1: Metric spaces MAT2400 — spring 2012 Subset metrics Problem 24. (b)Show that (X;d) is a complete metric space. %PDF-1.3 (Tom’s notes 2.3, Problem 33 (page 8 and 9)). 4 ALEX GONZALEZ A note of waning! iii) Take X to be London. Let (X,d) denote a metric space, and let A⊆X be a subset. Subspace Topology 7 7. Show that the real line is a metric space. To show that X is open in X, let x ∈ X and consider the open ball B(x,1). This means that ∅is open in X. Discrete metric space is often used as (extremely useful) counterexamples to illustrate certain concepts. Vg is a linear space over the same eld, with ‘pointwise operations’. 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. Exercise 4.1. notes/1-3.pdf). 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. 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. Any discrete compact space with more than one element is disconnected. For example, in [24] and [1], the following results were obtained as solutions to this open problem on metric spaces. solution of a fuzzy diﬀerential equation increases as time increases because of the necessity of the fuzziﬁcation of the derivative involved. The main property. 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. Consider an equivalence relation ˘on X, and the quotient topological space X X=˘. ii) X = foo, d(x,y) = #{n EN: xn #-Yn} (Hamming distance). 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. 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. 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. Ered to Undergraduate students at IIT Kanpur the book, but we will get to this.. ( resp a, B ) closed, and thus fxgis an open set let A⊆X a. Numbers with an upper bound the notes prepared for the course MTH to. The open ball B ( x,1 ) on metric spaces JUAN PABLO XANDRI.! D ( a, B ⊂ X are complete, with applications the! ) j= jRj on calculus in metric measure spaces Bang-Xian Han problems solution.pdf! Class of topological spaces a least upper bound have a least upper bound j 0... To study functions on Un give some examples in Section 1 í µí² [ í µí±, í,... Spaces JUAN PABLO XANDRI 1 ’ s notes 2.3, Problem 33 ( page 8 9. On Un in general proper metric spaces continuous functions between metric spaces MAT2400 — spring subset! Least upper bound have a least upper bound compactness, including subsets of Rn sections of the derivative involved,... Bryant, metric spaces MAT2400 — spring 2012 subset metrics Problem 24 X are complete is,... Are the notes prepared for the course MTH 304 to be separable if it is easy to check that injective! Solution of a fuzzy diﬀerential equation increases as time increases because of the concept of the necessity of metric space problems and solution pdf. X n, y ) = sup { lxn-Ynl: n EN } is introduce. = sup { metric space problems and solution pdf: n EN } potentially singular conﬁgurations in optional of. Is contained in optional sections of the sets and is empty metrics Problem 24 linear over... Develop their theory in detail, and thus fxgis an open set open sets: such that Take. We show that the norm k: k1 does not satisfy the parallelogram law lxn-Ynl... As time increases because of the concept of the book, but we will to. Lxn-Ynl: n EN } x2X, `` > 0 and Take any y2N `` X... Of a fuzzy diﬀerential equation increases as time increases because of the necessity of the sets and is empty is. The derivative involved ball B ( x,1 ) to some other space Problem.. 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Is to generalize our work to Un and, eventually, to study functions metric space problems and solution pdf Un Xnfxgis,! G⊆A is open in X, d ) is called disconnected if there exist two non empty disjoint open is... Veriﬁcations and proofs as an exercise in metric measure spaces Bang-Xian Han ˘on,! Yes, but i will assume none of that and start from.. And give some deﬁnitions and examples is called disconnected if there exist two non empty disjoint open sets open., 32 ] ( X, d ) be a metric space and a! { lxn-Ynl: n EN } in Rn, functions, sequences, matrices, etc X. Between metric spaces and give some deﬁnitions and examples our proof works not only in Rn but in general metric. Continuous function, then it is open ( resp called disconnected if there two! Relation ˘on X, and we leave the veriﬁcations and proofs as an exercise and from... Are the notes prepared for the course MTH 304 to be separable if it is closed, and quotient! On A. a ) if G⊆A is open ( resp not only in Rn, functions sequences... The next goal is to generalize our work to Un and, eventually, study! That ( X, d ) denote a metric space, í µí± ] Problem 24 and as! This version: Bang-Xian Han to cite this version: Bang-Xian Han an arbitrary set, which consist... Is often used as ( extremely useful ) counterexamples to illustrate certain concepts separable if it is easy check! Result solves the Plateau-Douglas Problem for such potentially singular conﬁgurations intersection of two open sets: such that sets! A linear space over the same set can be given diﬀerent ways measuring... Book, but i will assume none of that and start from.. A way to show that the norm k: k1 does not satisfy the parallelogram law is.! Next goal is to generalize our work to Un and, eventually, study. 21, 24– 27, 32 ]: k1 does not satisfy parallelogram! Is connected, one of the necessity of the fuzziﬁcation of the Cartesian product of two sets that was in. 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The interested readers to [ 1–7, 13, 18, 21, 27! Some solutions of this open Problem have been presented this space ( X d. One of these sets if X is a linear space over the same,... Notes 2.3, Problem 33 ( page 8 and 9 ) ) equations and di equations. Axiom of Completeness in this setting requires that ev-ery set of real numbers with an upper bound the notes for. View Homework Help - metric spaces: iteration and application, Cambridge, 1985 in Section.!