Problem Set 2: Solutions Math 201A: Fall 2016
|
|
- Daniella Jordan
- 5 years ago
- Views:
Transcription
1 Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that a compact subset of a metric space is closed and bounded. (a) If F X is closed and (x n ) is a Cauchy sequence in F, then (x n ) is Cauchy in X and x n x for some x X since X is complete. Then x F since F is closed, so F is complete. (b) Suppose that F X where F is closed and X is compact. If (x n ) is a sequence in F, then there is a subsequence (x nk ) that converges to x X since X is compact. Then x F since F is closed, so F is compact. Alternatively, If {G α X : α I} is an open cover of F, then {G α : α I} F c is an open cover of X. Since X is compact, there is a finite subcover of X which also covers F, so F is compact. (c) Let K X be compact. If (x n ) is a convergent sequence in K with limit x X, then every subsequence of (x n ) converges to x. Since K is compact, some subsequence of (x n ) converges to a limit in K, so x K and K is closed. Suppose that K is not bounded, and let x 1 K. Then for every r > 0 there exists x K such that d(x 1, x) r. Choose a sequence (x n ) in K as follows. Pick x 2 K such that d(x 1, x 2 ) 1. Given {x 1, x 2,..., x n }, pick x n+1 K such that By the triangle inequality, d(x 1, x n+1 ) 1 + max 1 k n d(x 1, x k ). d(x k, x n+1 ) d(x 1, x n+1 ) d(x 1, x k ) 1 for every 1 k n. It follows that d(x m, x n ) 1 for every m n, so (x n ) has no Cauchy subsequences, and therefore no convergent subsequences, so K is not compact. 1
2 Problem 2. Let A be a subset of a metric space X with closure Ā. Define the interior A and boundary A of A by A = {G A : G is open}, A = Ā \ A. (a) Why is A open and A closed? (b) Prove that X \ Ā = (X \ A). (c) Prove that A is closed if and only if A A, and A is open if and only if A A c. (d) If A is open, does it follow that (Ā) = A? (a) A union of open sets is open so A is open, and an intersection of closed sets is closed so Ā and A = Ā (A ) c are closed. (b) Note that x A if and only B ɛ (x) A for some ɛ > 0. If x Āc, then B ɛ (x) Āc for some ɛ > 0 since Āc is open. Since Ā A, we have Āc A c, so B ɛ (x) A c, meaning that x (A c ). It follows that Ā c (A c ). For the reverse inclusion, note that if x (A c ), then there exists ɛ > 0 such that B ɛ (x) A c, so x is not the limit of any sequence in A, meaning that x Āc. It follows that (A c ) Āc, so (A c ) = Āc. (c) If A is closed, then Ā = A so A = A (A ) c A. For the converse, note that since A A Ā, we have Ā = (Ā A ) (Ā (A ) c ) = A A. If A A, then it follows that closed. Ā A, so A = Ā, meaning that A is (d) This is not true in general. For example, define A R 2 by A = { (x, y) : x 2 + y 2 < 1 } \ {(x, 0) : 0 x < 1}. Then A is open, but (Ā) = {(x, y) : x 2 + y 2 < 1} A. 2
3 Problem 3. Let X be a metric space with a dense subset A X such that every Cauchy sequence in A converges in X. Prove that X is complete. Let (x n ) be a Cauchy sequence in X. Since A is dense in X, we can choose a sequence (a n ) in A such that d(x n, a n ) 0 as n. (For example, choose a n A such that d(x n, a n ) < 1/n.) Given any ɛ > 0, there exists M N such that d(x n, a n ) < ɛ/3 and d(x m, x n ) < ɛ/3 for all m, n > M, so d(a m, a n ) d(a m, x m ) + d(x m, x n ) + d(x n, a n ) < ɛ, which shows that (a n ) is a Cauchy sequence in A. It follows that (a n ) converges to some x X. Given any ɛ > 0, there exists N N such that d(x n, a n ) < ɛ/2 and d(a n, x) < ɛ/2 for all n > N, so d(x n, x) d(x n, a n ) + d(a n, x) < ɛ, which shows that (x n ) converges to x and proves that X is complete. 3
4 Problem 4. Let d : X X R be the discrete metric on a set X, { 1 if x y, d(x, y) = 0 if x = y. What are the compact subsets of the metric space (X, d)? A subset of X is compact if and only if it is finite. Every finite set is compact. If F = {x 1, x 2,..., x n } X and {G α X : α I} is an open cover of F, then x k G αk for some α k I, so {G α1, G α2,..., G αn } is a finite subcover of F. Alternatively, every sequence in F has a constant subsequence, which converges to a point in F. Conversely, if F X is infinite and G x = {x}, then {G x : x F } is an open cover of F with no finite subcover. Alternatively, if F is infinite, then there is a sequence (x n ) in F with x m x n for all m n, so d(x m, x n ) = 1 for m n, and (x n ) has no Cauchy or convergent subsequences. Remark. A rough heuristic is that compact sets have many properties in common with finite sets. For example, finite sets have the finite intersection property. 4
5 Problem 5. Let c 0 be the Banach space of real sequences (x n ) such that x n 0 as n with the sup-norm (x n ) = sup n N x n. Is the closed unit ball B = {(x n ) c 0 : (x n ) 1} compact? The closed unit ball in c 0 is not compact. For example, let e k = (δ nk ) n=1 δ nk = { 1 if n = k 0 if n k denote the sequence whose kth term is one and whose other terms are zero. Then e k c 0 since lim n δ nk = 0, and e k belongs to the closed unit ball in c 0 since e k = 1. However, e j e k = 1 for every j k, so (e k ) k=1 has no Cauchy or convergent subsequences in c 0. Remark. A similar argument using the Riesz lemma shows that the closed unit ball in any infinite-dimensional normed space is not compact in the norm topology. 5
6 Problem 6. A metric (or topological) space X is disconnected if there are non-empty open sets U, V X such that X = U V and U V =. A space is connected if it is not disconnected. A space X is totally disconnected if its only non-empty connected subsets are the singleton sets {x} with x X. (a) Show that the interval [0, 1] is connected (in its standard metric topology). (b) Show that the set Q of rational numbers is totally disconnected. (a) Suppose for contradiction that [0, 1] = U V where U, V are nonempty, disjoint open sets in [0, 1]. We assume that 0 U without loss of generality. Let a = sup {x [0, 1] : [0, x) U}. Since 0 U and U is open, we have [0, ɛ) U for some ɛ > 0, so 0 < a 1. If 0 < b < a, then [0, b) U since, by the definition of the supremum, there exists b < c < a such that [0, c) U. (It also follows that [0, a) = 0<b<a [0, b) U, so the supremum is attained and, in fact, {x [0, 1] : [0, x) U} = [0, a].) If a U, then (a ɛ, a + ɛ) U for some ɛ > 0, but then [0, a + ɛ) U, contradicting the definition of a. On the other hand, if a V, then (a ɛ, a] V for some 0 < ɛ < a, but then [0, b) U for a ɛ < b a, also contradicting the definition of a. (b) Let A Q be any subset of the rational numbers with at least two elements. Choose x, y A with x y. The irrational numbers are dense in R, so there exists z R \ Q such that x < z < y. Let U = (, z) A, V = (z, ) A. Then U, V are open sets in the relative topology on A. Moreover, x U, y V so U, V are nonempty, and U V =, U V = A. It follows that Q is totally disconnected. 6
7 Problem 7. Let T = {G R : G c is countable or G c = R}. (a) Show that T is a topology on R. (b) Let I = (0, 1) with closure Ī = {F I : F is closed} in this topology. Show that Ī = R. (c) Is there a sequence (x n ) such that x n I and x n 2 in this topology? (d) What part of your proof in Problem 5 of Set 1 fails in this example? (a) The collection of closed sets C = {F R : F c T } in this topology is given by C = {F R : F is countable or F = R}. The collection C satisfies the axioms for closed sets in a topological space: (1), R C. (2) The intersection of closed sets is closed, since either every set is R and the intersection is R, or at least one set is countable and the intersection in countable, since any subset of a countable set is countable. (3) A finite union of closed sets is closed, since a finite (or countable) union of countable sets is countable. It follows that T is a topology on R (called the co-countable topology). (b) If F I is closed, then F is uncountable, since (0, 1) is uncountable, so F = R and Ī = R. (c) Let (x n ) n=1 be a sequence in (0, 1). Then U = R \ {x n : n N} is an open neighborhood of 2, but x n / U for any n N, so (x n ) does not converge to 2. A similar argument applies to any x / (0, 1), so the sequential closure of I is Ĩ = (0, 1). (d) If X is a topological space, then a neighborhood base of x X is a collection {U α : α A} of neighborhoods of x such that for every neighborhood U of x there exists α A with U α U. Then x n x if and only if for every α A there exists N N such that x n U α for all n > N. The proof that the sequential closure is equal to the closure fails for the co-countable topology on R because x R does not have a countable neighborhood base. On the other hand, if X is a metric space, then {B 1/n (x) : n N} is a countable neighborhood base of any x X. 7
8 Problem 8. Let S be the set of sequences whose terms are 0 or 1: S = {(s k ) k=1 : s k = 0 or s k = 1}. (a) Use a diagonal argument to show that S is uncountable. (b) Show that S has the same cardinality as the power set P(N) of the natural numbers. (a) Let f : N S. Suppose that f(n) = s n where s n = (s kn ) k=1, and define t = (t k ) k=1 S by { 1 if s kk = 0, t k = 0 if s kk = 1. Then t s k for every k N, since the two sequences have different kth terms. It follows that there is no map from N onto S, so S is uncountably infinite. For A N define the characteristic function χ A : N {0, 1} of A by { 1 if k A, χ A (k) = 0 if k / A. is one-to- Then the map f : P(N) S defined by f(a) = (χ A (k)) k=1 one and onto, so P(N) and S have the same cardinality. 8
1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3
Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,
More informationREVIEW OF ESSENTIAL MATH 346 TOPICS
REVIEW OF ESSENTIAL MATH 346 TOPICS 1. AXIOMATIC STRUCTURE OF R Doğan Çömez The real number system is a complete ordered field, i.e., it is a set R which is endowed with addition and multiplication operations
More informationMATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION. Problem 1
MATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION ELEMENTS OF SOLUTION Problem 1 1. Let X be a Hausdorff space and K 1, K 2 disjoint compact subsets of X. Prove that there exist disjoint open sets U 1 and
More informationReal Analysis Notes. Thomas Goller
Real Analysis Notes Thomas Goller September 4, 2011 Contents 1 Abstract Measure Spaces 2 1.1 Basic Definitions........................... 2 1.2 Measurable Functions........................ 2 1.3 Integration..............................
More informationMATH 202B - Problem Set 5
MATH 202B - Problem Set 5 Walid Krichene (23265217) March 6, 2013 (5.1) Show that there exists a continuous function F : [0, 1] R which is monotonic on no interval of positive length. proof We know there
More informationMath 320-2: Midterm 2 Practice Solutions Northwestern University, Winter 2015
Math 30-: Midterm Practice Solutions Northwestern University, Winter 015 1. Give an example of each of the following. No justification is needed. (a) A metric on R with respect to which R is bounded. (b)
More informationMAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9
MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended
More informationMid Term-1 : Practice problems
Mid Term-1 : Practice problems These problems are meant only to provide practice; they do not necessarily reflect the difficulty level of the problems in the exam. The actual exam problems are likely to
More informationMH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then
MH 7500 THEOREMS Definition. A topological space is an ordered pair (X, T ), where X is a set and T is a collection of subsets of X such that (i) T and X T ; (ii) U V T whenever U, V T ; (iii) U T whenever
More informationContents Ordered Fields... 2 Ordered sets and fields... 2 Construction of the Reals 1: Dedekind Cuts... 2 Metric Spaces... 3
Analysis Math Notes Study Guide Real Analysis Contents Ordered Fields 2 Ordered sets and fields 2 Construction of the Reals 1: Dedekind Cuts 2 Metric Spaces 3 Metric Spaces 3 Definitions 4 Separability
More informationReal Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi
Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.
More informationHW 4 SOLUTIONS. , x + x x 1 ) 2
HW 4 SOLUTIONS The Way of Analysis p. 98: 1.) Suppose that A is open. Show that A minus a finite set is still open. This follows by induction as long as A minus one point x is still open. To see that A
More information1 Lecture 4: Set topology on metric spaces, 8/17/2012
Summer Jump-Start Program for Analysis, 01 Song-Ying Li 1 Lecture : Set topology on metric spaces, 8/17/01 Definition 1.1. Let (X, d) be a metric space; E is a subset of X. Then: (i) x E is an interior
More informationChapter 2 Metric Spaces
Chapter 2 Metric Spaces The purpose of this chapter is to present a summary of some basic properties of metric and topological spaces that play an important role in the main body of the book. 2.1 Metrics
More informationMaths 212: Homework Solutions
Maths 212: Homework Solutions 1. The definition of A ensures that x π for all x A, so π is an upper bound of A. To show it is the least upper bound, suppose x < π and consider two cases. If x < 1, then
More informationExam 2 extra practice problems
Exam 2 extra practice problems (1) If (X, d) is connected and f : X R is a continuous function such that f(x) = 1 for all x X, show that f must be constant. Solution: Since f(x) = 1 for every x X, either
More informationMath 4317 : Real Analysis I Mid-Term Exam 1 25 September 2012
Instructions: Answer all of the problems. Math 4317 : Real Analysis I Mid-Term Exam 1 25 September 2012 Definitions (2 points each) 1. State the definition of a metric space. A metric space (X, d) is set
More informationIn N we can do addition, but in order to do subtraction we need to extend N to the integers
Chapter The Real Numbers.. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {, 2, 3, }. In N we can do addition, but in order to do subtraction we need to extend
More information7 Complete metric spaces and function spaces
7 Complete metric spaces and function spaces 7.1 Completeness Let (X, d) be a metric space. Definition 7.1. A sequence (x n ) n N in X is a Cauchy sequence if for any ɛ > 0, there is N N such that n, m
More informationSolutions to Tutorial 8 (Week 9)
The University of Sydney School of Mathematics and Statistics Solutions to Tutorial 8 (Week 9) MATH3961: Metric Spaces (Advanced) Semester 1, 2018 Web Page: http://www.maths.usyd.edu.au/u/ug/sm/math3961/
More informationEconomics 204 Fall 2012 Problem Set 3 Suggested Solutions
Economics 204 Fall 2012 Problem Set 3 Suggested Solutions 1. Give an example of each of the following (and prove that your example indeed works): (a) A complete metric space that is bounded but not compact.
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets
FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES CHRISTOPHER HEIL 1. Compact Sets Definition 1.1 (Compact and Totally Bounded Sets). Let X be a metric space, and let E X be
More informationIn N we can do addition, but in order to do subtraction we need to extend N to the integers
Chapter 1 The Real Numbers 1.1. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {1, 2, 3, }. In N we can do addition, but in order to do subtraction we need
More informationAnalysis Finite and Infinite Sets The Real Numbers The Cantor Set
Analysis Finite and Infinite Sets Definition. An initial segment is {n N n n 0 }. Definition. A finite set can be put into one-to-one correspondence with an initial segment. The empty set is also considered
More informationMath 117: Topology of the Real Numbers
Math 117: Topology of the Real Numbers John Douglas Moore November 10, 2008 The goal of these notes is to highlight the most important topics presented in Chapter 3 of the text [1] and to provide a few
More informationAnalysis Comprehensive Exam Questions Fall F(x) = 1 x. f(t)dt. t 1 2. tf 2 (t)dt. and g(t, x) = 2 t. 2 t
Analysis Comprehensive Exam Questions Fall 2. Let f L 2 (, ) be given. (a) Prove that ( x 2 f(t) dt) 2 x x t f(t) 2 dt. (b) Given part (a), prove that F L 2 (, ) 2 f L 2 (, ), where F(x) = x (a) Using
More information4 Countability axioms
4 COUNTABILITY AXIOMS 4 Countability axioms Definition 4.1. Let X be a topological space X is said to be first countable if for any x X, there is a countable basis for the neighborhoods of x. X is said
More informationPart III. 10 Topological Space Basics. Topological Spaces
Part III 10 Topological Space Basics Topological Spaces Using the metric space results above as motivation we will axiomatize the notion of being an open set to more general settings. Definition 10.1.
More informationMath 426 Homework 4 Due 3 November 2017
Math 46 Homework 4 Due 3 November 017 1. Given a metric space X,d) and two subsets A,B, we define the distance between them, dista,b), as the infimum inf a A, b B da,b). a) Prove that if A is compact and
More informationProblem Set 1: Solutions Math 201A: Fall Problem 1. Let (X, d) be a metric space. (a) Prove the reverse triangle inequality: for every x, y, z X
Problem Set 1: s Math 201A: Fall 2016 Problem 1. Let (X, d) be a metric space. (a) Prove the reverse triangle inequality: for every x, y, z X d(x, y) d(x, z) d(z, y). (b) Prove that if x n x and y n y
More informationMAS3706 Topology. Revision Lectures, May I do not answer enquiries as to what material will be in the exam.
MAS3706 Topology Revision Lectures, May 208 Z.A.Lykova It is essential that you read and try to understand the lecture notes from the beginning to the end. Many questions from the exam paper will be similar
More informationLecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University
Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University February 7, 2007 2 Contents 1 Metric Spaces 1 1.1 Basic definitions...........................
More informationSome Background Material
Chapter 1 Some Background Material In the first chapter, we present a quick review of elementary - but important - material as a way of dipping our toes in the water. This chapter also introduces important
More informationTopology. Xiaolong Han. Department of Mathematics, California State University, Northridge, CA 91330, USA address:
Topology Xiaolong Han Department of Mathematics, California State University, Northridge, CA 91330, USA E-mail address: Xiaolong.Han@csun.edu Remark. You are entitled to a reward of 1 point toward a homework
More informationMath General Topology Fall 2012 Homework 13 Solutions
Math 535 - General Topology Fall 2012 Homework 13 Solutions Note: In this problem set, function spaces are endowed with the compact-open topology unless otherwise noted. Problem 1. Let X be a compact topological
More informationSOLUTIONS TO SOME PROBLEMS
23 FUNCTIONAL ANALYSIS Spring 23 SOLUTIONS TO SOME PROBLEMS Warning:These solutions may contain errors!! PREPARED BY SULEYMAN ULUSOY PROBLEM 1. Prove that a necessary and sufficient condition that the
More informationREAL AND COMPLEX ANALYSIS
REAL AND COMPLE ANALYSIS Third Edition Walter Rudin Professor of Mathematics University of Wisconsin, Madison Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any
More informationCLASS NOTES FOR APRIL 14, 2000
CLASS NOTES FOR APRIL 14, 2000 Announcement: Section 1.2, Questions 3,5 have been deferred from Assignment 1 to Assignment 2. Section 1.4, Question 5 has been dropped entirely. 1. Review of Wednesday class
More informationIntroduction to Proofs in Analysis. updated December 5, By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION
Introduction to Proofs in Analysis updated December 5, 2016 By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION Purpose. These notes intend to introduce four main notions from
More informationProblem Set 5. 2 n k. Then a nk (x) = 1+( 1)k
Problem Set 5 1. (Folland 2.43) For x [, 1), let 1 a n (x)2 n (a n (x) = or 1) be the base-2 expansion of x. (If x is a dyadic rational, choose the expansion such that a n (x) = for large n.) Then the
More informationMath 5210, Definitions and Theorems on Metric Spaces
Math 5210, Definitions and Theorems on Metric Spaces Let (X, d) be a metric space. We will use the following definitions (see Rudin, chap 2, particularly 2.18) 1. Let p X and r R, r > 0, The ball of radius
More informationFunctional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...
Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................
More informationconverges as well if x < 1. 1 x n x n 1 1 = 2 a nx n
Solve the following 6 problems. 1. Prove that if series n=1 a nx n converges for all x such that x < 1, then the series n=1 a n xn 1 x converges as well if x < 1. n For x < 1, x n 0 as n, so there exists
More information1. Let A R be a nonempty set that is bounded from above, and let a be the least upper bound of A. Show that there exists a sequence {a n } n N
Applied Analysis prelim July 15, 216, with solutions Solve 4 of the problems 1-5 and 2 of the problems 6-8. We will only grade the first 4 problems attempted from1-5 and the first 2 attempted from problems
More informationABSTRACT INTEGRATION CHAPTER ONE
CHAPTER ONE ABSTRACT INTEGRATION Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any form or by any means. Suggestions and errors are invited and can be mailed
More informationProblem set 1, Real Analysis I, Spring, 2015.
Problem set 1, Real Analysis I, Spring, 015. (1) Let f n : D R be a sequence of functions with domain D R n. Recall that f n f uniformly if and only if for all ɛ > 0, there is an N = N(ɛ) so that if n
More informationAnalysis III Theorems, Propositions & Lemmas... Oh My!
Analysis III Theorems, Propositions & Lemmas... Oh My! Rob Gibson October 25, 2010 Proposition 1. If x = (x 1, x 2,...), y = (y 1, y 2,...), then is a distance. ( d(x, y) = x k y k p Proposition 2. In
More informationNAME: Mathematics 205A, Fall 2008, Final Examination. Answer Key
NAME: Mathematics 205A, Fall 2008, Final Examination Answer Key 1 1. [25 points] Let X be a set with 2 or more elements. Show that there are topologies U and V on X such that the identity map J : (X, U)
More informationvan Rooij, Schikhof: A Second Course on Real Functions
vanrooijschikhofproblems.tex December 5, 2017 http://thales.doa.fmph.uniba.sk/sleziak/texty/rozne/pozn/books/ van Rooij, Schikhof: A Second Course on Real Functions Some notes made when reading [vrs].
More informationA LITTLE REAL ANALYSIS AND TOPOLOGY
A LITTLE REAL ANALYSIS AND TOPOLOGY 1. NOTATION Before we begin some notational definitions are useful. (1) Z = {, 3, 2, 1, 0, 1, 2, 3, }is the set of integers. (2) Q = { a b : aεz, bεz {0}} is the set
More informationTHEOREMS, ETC., FOR MATH 515
THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every
More informationLecture 5 - Hausdorff and Gromov-Hausdorff Distance
Lecture 5 - Hausdorff and Gromov-Hausdorff Distance August 1, 2011 1 Definition and Basic Properties Given a metric space X, the set of closed sets of X supports a metric, the Hausdorff metric. If A is
More informationMetric Spaces and Topology
Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies
More informationReal Analysis Chapter 4 Solutions Jonathan Conder
2. Let x, y X and suppose that x y. Then {x} c is open in the cofinite topology and contains y but not x. The cofinite topology on X is therefore T 1. Since X is infinite it contains two distinct points
More informationMATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous:
MATH 51H Section 4 October 16, 2015 1 Continuity Recall what it means for a function between metric spaces to be continuous: Definition. Let (X, d X ), (Y, d Y ) be metric spaces. A function f : X Y is
More informationANALYSIS WORKSHEET II: METRIC SPACES
ANALYSIS WORKSHEET II: METRIC SPACES Definition 1. A metric space (X, d) is a space X of objects (called points), together with a distance function or metric d : X X [0, ), which associates to each pair
More informationSets, Functions and Metric Spaces
Chapter 14 Sets, Functions and Metric Spaces 14.1 Functions and sets 14.1.1 The function concept Definition 14.1 Let us consider two sets A and B whose elements may be any objects whatsoever. Suppose that
More informationTopological properties
CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological
More informationMTG 5316/4302 FALL 2018 REVIEW FINAL
MTG 5316/4302 FALL 2018 REVIEW FINAL JAMES KEESLING Problem 1. Define open set in a metric space X. Define what it means for a set A X to be connected in a metric space X. Problem 2. Show that if a set
More informationMetric Spaces Math 413 Honors Project
Metric Spaces Math 413 Honors Project 1 Metric Spaces Definition 1.1 Let X be a set. A metric on X is a function d : X X R such that for all x, y, z X: i) d(x, y) = d(y, x); ii) d(x, y) = 0 if and only
More informationFunctional Analysis Exercise Class
Functional Analysis Exercise Class Week 2 November 6 November Deadline to hand in the homeworks: your exercise class on week 9 November 13 November Exercises (1) Let X be the following space of piecewise
More informationMA651 Topology. Lecture 9. Compactness 2.
MA651 Topology. Lecture 9. Compactness 2. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology
More informationIntroduction to Topology
Chapter 2 Introduction to Topology In this chapter, we will use the tools we developed concerning sequences and series to study two other mathematical objects: sets and functions. For definitions of set
More informationPart A. Metric and Topological Spaces
Part A Metric and Topological Spaces 1 Lecture A1 Introduction to the Course This course is an introduction to the basic ideas of topology and metric space theory for first-year graduate students. Topology
More information2.31 Definition By an open cover of a set E in a metric space X we mean a collection {G α } of open subsets of X such that E α G α.
Chapter 2. Basic Topology. 2.3 Compact Sets. 2.31 Definition By an open cover of a set E in a metric space X we mean a collection {G α } of open subsets of X such that E α G α. 2.32 Definition A subset
More informationSpaces of continuous functions
Chapter 2 Spaces of continuous functions 2.8 Baire s Category Theorem Recall that a subset A of a metric space (X, d) is dense if for all x X there is a sequence from A converging to x. An equivalent definition
More informationReal Analysis. July 10, These notes are intended for use in the warm-up camp for incoming Berkeley Statistics
Real Analysis July 10, 2006 1 Introduction These notes are intended for use in the warm-up camp for incoming Berkeley Statistics graduate students. Welcome to Cal! The real analysis review presented here
More informationWeek 2: Sequences and Series
QF0: Quantitative Finance August 29, 207 Week 2: Sequences and Series Facilitator: Christopher Ting AY 207/208 Mathematicians have tried in vain to this day to discover some order in the sequence of prime
More informationEconomics 204 Fall 2011 Problem Set 2 Suggested Solutions
Economics 24 Fall 211 Problem Set 2 Suggested Solutions 1. Determine whether the following sets are open, closed, both or neither under the topology induced by the usual metric. (Hint: think about limit
More informationMAT 544 Problem Set 2 Solutions
MAT 544 Problem Set 2 Solutions Problems. Problem 1 A metric space is separable if it contains a dense subset which is finite or countably infinite. Prove that every totally bounded metric space X is separable.
More informationChapter 3: Baire category and open mapping theorems
MA3421 2016 17 Chapter 3: Baire category and open mapping theorems A number of the major results rely on completeness via the Baire category theorem. 3.1 The Baire category theorem 3.1.1 Definition. A
More informationThis chapter contains a very bare summary of some basic facts from topology.
Chapter 2 Topological Spaces This chapter contains a very bare summary of some basic facts from topology. 2.1 Definition of Topology A topology O on a set X is a collection of subsets of X satisfying the
More information11691 Review Guideline Real Analysis. Real Analysis. - According to Principles of Mathematical Analysis by Walter Rudin (Chapter 1-4)
Real Analysis - According to Principles of Mathematical Analysis by Walter Rudin (Chapter 1-4) 1 The Real and Complex Number Set: a collection of objects. Proper subset: if A B, then call A a proper subset
More informationM17 MAT25-21 HOMEWORK 6
M17 MAT25-21 HOMEWORK 6 DUE 10:00AM WEDNESDAY SEPTEMBER 13TH 1. To Hand In Double Series. The exercises in this section will guide you to complete the proof of the following theorem: Theorem 1: Absolute
More informationTheorems. Theorem 1.11: Greatest-Lower-Bound Property. Theorem 1.20: The Archimedean property of. Theorem 1.21: -th Root of Real Numbers
Page 1 Theorems Wednesday, May 9, 2018 12:53 AM Theorem 1.11: Greatest-Lower-Bound Property Suppose is an ordered set with the least-upper-bound property Suppose, and is bounded below be the set of lower
More informationWalker Ray Econ 204 Problem Set 3 Suggested Solutions August 6, 2015
Problem 1. Take any mapping f from a metric space X into a metric space Y. Prove that f is continuous if and only if f(a) f(a). (Hint: use the closed set characterization of continuity). I make use of
More informationFrom now on, we will represent a metric space with (X, d). Here are some examples: i=1 (x i y i ) p ) 1 p, p 1.
Chapter 1 Metric spaces 1.1 Metric and convergence We will begin with some basic concepts. Definition 1.1. (Metric space) Metric space is a set X, with a metric satisfying: 1. d(x, y) 0, d(x, y) = 0 x
More informationg 2 (x) (1/3)M 1 = (1/3)(2/3)M.
COMPACTNESS If C R n is closed and bounded, then by B-W it is sequentially compact: any sequence of points in C has a subsequence converging to a point in C Conversely, any sequentially compact C R n is
More informationBased on the Appendix to B. Hasselblatt and A. Katok, A First Course in Dynamics, Cambridge University press,
NOTE ON ABSTRACT RIEMANN INTEGRAL Based on the Appendix to B. Hasselblatt and A. Katok, A First Course in Dynamics, Cambridge University press, 2003. a. Definitions. 1. Metric spaces DEFINITION 1.1. If
More informationYour first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
More informationMath General Topology Fall 2012 Homework 11 Solutions
Math 535 - General Topology Fall 2012 Homework 11 Solutions Problem 1. Let X be a topological space. a. Show that the following properties of a subset A X are equivalent. 1. The closure of A in X has empty
More informationIntroductory Analysis I Fall 2014 Homework #5 Solutions
Introductory Analysis I Fall 2014 Homework #5 Solutions 6. Let M be a metric space, let C D M. Now we can think of C as a subset of the metric space M or as a subspace of the metric space D (D being a
More informationMeasures and Measure Spaces
Chapter 2 Measures and Measure Spaces In summarizing the flaws of the Riemann integral we can focus on two main points: 1) Many nice functions are not Riemann integrable. 2) The Riemann integral does not
More informationSolve EACH of the exercises 1-3
Topology Ph.D. Entrance Exam, August 2011 Write a solution of each exercise on a separate page. Solve EACH of the exercises 1-3 Ex. 1. Let X and Y be Hausdorff topological spaces and let f: X Y be continuous.
More informationSequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1.
Chapter 3 Sequences Both the main elements of calculus (differentiation and integration) require the notion of a limit. Sequences will play a central role when we work with limits. Definition 3.. A Sequence
More informationMath General Topology Fall 2012 Homework 8 Solutions
Math 535 - General Topology Fall 2012 Homework 8 Solutions Problem 1. (Willard Exercise 19B.1) Show that the one-point compactification of R n is homeomorphic to the n-dimensional sphere S n. Note that
More informationThat is, there is an element
Section 3.1: Mathematical Induction Let N denote the set of natural numbers (positive integers). N = {1, 2, 3, 4, } Axiom: If S is a nonempty subset of N, then S has a least element. That is, there is
More informationHonours Analysis III
Honours Analysis III Math 354 Prof. Dmitry Jacobson Notes Taken By: R. Gibson Fall 2010 1 Contents 1 Overview 3 1.1 p-adic Distance............................................ 4 2 Introduction 5 2.1 Normed
More informationLocally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem
56 Chapter 7 Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem Recall that C(X) is not a normed linear space when X is not compact. On the other hand we could use semi
More informationl(y j ) = 0 for all y j (1)
Problem 1. The closed linear span of a subset {y j } of a normed vector space is defined as the intersection of all closed subspaces containing all y j and thus the smallest such subspace. 1 Show that
More informationMetric Spaces Math 413 Honors Project
Metric Spaces Math 413 Honors Project 1 Metric Spaces Definition 1.1 Let X be a set. A metric on X is a function d : X X R such that for all x, y, z X: i) d(x, y) = d(y, x); ii) d(x, y) = 0 if and only
More informationContinuous Functions on Metric Spaces
Continuous Functions on Metric Spaces Math 201A, Fall 2016 1 Continuous functions Definition 1. Let (X, d X ) and (Y, d Y ) be metric spaces. A function f : X Y is continuous at a X if for every ɛ > 0
More informationMATH 4200 HW: PROBLEM SET FOUR: METRIC SPACES
MATH 4200 HW: PROBLEM SET FOUR: METRIC SPACES PETE L. CLARK 4. Metric Spaces (no more lulz) Directions: This week, please solve any seven problems. Next week, please solve seven more. Starred parts of
More informationReal Analysis, 2nd Edition, G.B.Folland Elements of Functional Analysis
Real Analysis, 2nd Edition, G.B.Folland Chapter 5 Elements of Functional Analysis Yung-Hsiang Huang 5.1 Normed Vector Spaces 1. Note for any x, y X and a, b K, x+y x + y and by ax b y x + b a x. 2. It
More informationReal Analysis Problems
Real Analysis Problems Cristian E. Gutiérrez September 14, 29 1 1 CONTINUITY 1 Continuity Problem 1.1 Let r n be the sequence of rational numbers and Prove that f(x) = 1. f is continuous on the irrationals.
More informationMathematics 220 Workshop Cardinality. Some harder problems on cardinality.
Some harder problems on cardinality. These are two series of problems with specific goals: the first goal is to prove that the cardinality of the set of irrational numbers is continuum, and the second
More informationECARES Université Libre de Bruxelles MATH CAMP Basic Topology
ECARES Université Libre de Bruxelles MATH CAMP 03 Basic Topology Marjorie Gassner Contents: - Subsets, Cartesian products, de Morgan laws - Ordered sets, bounds, supremum, infimum - Functions, image, preimage,
More informationReal Analysis - Notes and After Notes Fall 2008
Real Analysis - Notes and After Notes Fall 2008 October 29, 2008 1 Introduction into proof August 20, 2008 First we will go through some simple proofs to learn how one writes a rigorous proof. Let start
More informationLogical Connectives and Quantifiers
Chapter 1 Logical Connectives and Quantifiers 1.1 Logical Connectives 1.2 Quantifiers 1.3 Techniques of Proof: I 1.4 Techniques of Proof: II Theorem 1. Let f be a continuous function. If 1 f(x)dx 0, then
More informationTopology Part of the Qualify Exams of Department of Mathematics, Texas A&M University Prepared by Zhang, Zecheng
Topology Part of the Qualify Exams of Department of Mathematics, Texas A&M University Prepared by Zhang, Zecheng Remark 0.1. This is a solution Manuel to the topology questions of the Topology Geometry
More information