Topology Homework Assignment 1 Solutions
|
|
- Arleen Conley
- 6 years ago
- Views:
Transcription
1 Topology Homework Assignment 1 Solutions 1. Prove that R n with the usual topology satisfies the axioms for a topological space. Let U denote the usual topology on R n. 1(a) R n U because if x R n, then B 1 (x) R n. 1(b) U because if x, then B 1 (x) R n. 2. Suppose that A is an index set (possibly infinite), and Then if V = {V α : α A} U. x V = α A V α, there is some α 0 A, such that x V α0. Since V α0 is open, there is a real number r > 0, such that B r (x) V α0. But then and hence V is open. 3. Suppose that and B r (x) α A V α = V, {V 1, V 2,... V k } U, x k i=1v i. Since each V i is open, for each i = 1,... k, there is a real number r i such that B ri (x) V i. 1
2 Furthermore, since B ri (x) B rj (x) when r i < r j, if r = min{r i : i = 1,..., k}, for i = 1,... k. Hence and k i=1v i, is open. B r (x) B ri (x) V i, B r (x) k i=1v i, 2. Prove that the ɛ δ definition of continuity is equivalent to the open set definition of continuity (i.e., f : R n R m is continuous iff f 1 (V ) is open for each open subset V R m ). We first show that f is continuous at x 0 R n in the ɛ δ definition iff for each open set V containing f(x 0 ), there is an open set U containing x 0 such that f(u) V. So suppose first that f is continuous at x 0 in the ɛ δ definition, and let V be an open set containing f(x 0 ). Since R m has the usual topology, there exists a real number ɛ > 0 such that B ɛ [f(x 0 )] V. Since we re assuming f is continuous at x 0 in the ɛ δ definition, there is some real number δ > 0 such that But and x x 0 < δ f(x) f(x 0 ) < ɛ. {x R n : x x 0 < δ} = B δ (x 0 ), {y R n : y f(x 0 ) < ɛ} = B ɛ [f(x 0 )]. So if x B δ (x 0 ), then f(x) B ɛ [f(x 0 )]. Since B ɛ [f(x 0 )] V, we have that x B δ (x 0 ) f(x) V. That is, f[b δ (x 0 )] V. Since B δ (x 0 ) is open, we can let U = B δ (x 0 ). 2
3 Conversely, suppose that for each open set V containing f(x 0 ), there is an open set U containing x 0 such that f(u) V. Let ɛ > 0. Then B ɛ [f(x 0 )] is an open set containing f(x 0 ). Hence there is an open set U containing x 0 such that f(u) V. Since R n has the usual topology, there is a δ > 0, such that B δ (x 0 ) U. But then f[b δ (x 0 )] f(u) B ɛ [f(x 0 )]. Since and we have that B δ (x 0 ) = {x R n : x x 0 < δ}, B ɛ [f(x 0 )] = {y R n : y f(x 0 ) < ɛ}, x x 0 < δ f(x) f(x 0 ) < ɛ. Now suppose that f is continuous at each point x of R n in the ɛ δ definition. We want to show that if V is an open subset of R m, then f 1 (V ) is an open subset of R n. Let V be an open subset of R m. Then for each f(x) V, by our preceding result, there is a δ x > 0, such that But then f[b δx (x)] V. f 1 (V ) = x f 1 (V )B δx (x) because every x f 1 (V ) belongs to the right-hand side, and every B δx is contained in the left-hand side. Since f 1 (V ) is a union of open sets, f 1 (V ) is open. Conversely suppose that for each open subset V R m, f 1 (V ) is open in R n, and let x 0 R n and ɛ > 0. Then B ɛ [f(x 0 )] is an open subset of R m. So f 1 {B ɛ [f(x 0 )]} is an open subset of R n. Thus, since x f 1 {B ɛ [f(x 0 )]}, there must exist an open ball B δ (x 0 ) f 1 {B ɛ [f(x 0 )]}. Thus if x x 0 < δ, (i.e., if x B δ (x 0 ), then f(x) f(x 0 ) < ɛ since f(x) B ɛ [f(x 0 ). 3
4 3. Suppose (X, U) is a topological space and Y X. Define a family of subsets V of Y by V V iff there exists a set U U such that V = U Y. Prove that V is a topology on Y. V is called the relative topology on Y, and unless we state otherwise, you should assume that a subset of a topological space has the relative topology. 1(a) Since Y X, Y = Y X, and since X is an element of U, Y is open in the relative topology. 1(b) Since = Y, and since is an element of U, is open in the relative topology. 2. Let A be an index set and let V = {V α : α A} be a family of elements of the relative topology. Then for each element V α V, there is an open set U α U, such that Thus V α = U α Y. α A V α = α A (U α Y ) = ( α A U α ) Y. But U α is open in X. So V α = ( U α ) Y is open in the relative topology. 3. Let {V 1, V 2,... V k } be a finite subset of the relative topology. Then for i = 1,... k, there is an element U i U, such that V i = U i Y. Thus k i=1v i = k i=1(u i Y ) = ( k i=1u i ) Y. Since U i is open in X, V i = ( U i ) Y is open in the relative topology. 4. Using subsets of euclidean spaces, find topological spaces X and Y and a function f : X Y, such that (a) f is continuous but not open. (b) f is open but not continuous. 4
5 Can you find a function f satisfying 4a or 4b that is both one-to-one and onto? Let f : [0, 2π) S 1 by f(t) = (cos(t), sin(t)). Then f is both one-toone and onto, for if (x, y) S 1, we can rewrite (x, y) in exponential form e it, where t is a unique value in [0, 2π), and f(t) = (x, y). Thus, f has a unique inverse g : S 1 [0, 2π). Since the component functions, cos(t) and sin(t), are continuous, f is continuous. (Technically, we need to know about product spaces in order to prove this result.) However, f is not open. For open subsets of S 1 (in the relative topology on S 1 as a subset of R 2 ) are unions of curved open intervals. To see this observe that any open subset of R 2 is a union of open balls. Since an open ball intersects S 1 in a curved open interval (or all of S 1 ), any open subset of S 1 is a union of open intervals. But open sets in [0, 2π) are unions of intersections of open intervals with [0, 2π). Thus, [0, π) is open in [0, 2π), and f([0, π)) = {(x, y) S 1 : y > 0} {(1, 0)}, which isn t a curved open interval, and hence isn t open. The function g, on the other hand is open, but it s not continuous. Since f is continuous, if U is open in S 1, f 1 (U) = g(u) is open in [0, 2π). However, g 1 ([0, π)) = f([0, π)) isn t open. So g is not continuous. 5. Suppose X and Y are topological spaces, and f : X Y is continuous. If A is a compact subset of X, show that f(a) is a compact subset of Y. (Use the open cover definition of compactness.) So compactness is a topological property: if Y is homeomorphic to X (denoted Y X), and X is compact, then so is Y. We proved this in class. 6. Suppose X is a topological space. Show that a subset C of X is closed iff C contains all of the points that are arbitrarily close to C. (Recall that the point x is arbitrarily close to C if every open neighborhood of x contains a point of C.) We proved this in class. 5
6 7. Suppose that X and Y are topological spaces and f : X Y. Show that f is continuous iff for each closed set C Y, f 1 (C) is closed a closed subset of X. Suppose first that f is continuous i.e., for each open set V in Y, f 1 (V ) is open in X. Let C be a closed subset of Y. Then Y C is open in Y. Furthermore, f 1 (Y C) = X f 1 (C), by Theorem 2.6 on page 21 of the set theory handout. But, by assumption f is continuous. So f 1 (Y C) is open in X. Thus, is closed in X. X [X f 1 (C)] = f 1 (C) Conversely, suppose that for each closed set C in Y, f 1 (C) is closed in in X. Also suppose that V is an open subset of Y. Then, by assumption, is closed, and hence is open. That is, f is continuous. f 1 (Y V ) = X f 1 (V ) X [X f 1 (V )] = f 1 (V ) 6
1 The Local-to-Global Lemma
Point-Set Topology Connectedness: Lecture 2 1 The Local-to-Global Lemma In the world of advanced mathematics, we are often interested in comparing the local properties of a space to its global properties.
More informationMath General Topology Fall 2012 Homework 1 Solutions
Math 535 - General Topology Fall 2012 Homework 1 Solutions Definition. Let V be a (real or complex) vector space. A norm on V is a function : V R satisfying: 1. Positivity: x 0 for all x V and moreover
More informationProblem Set 2: Solutions Math 201A: Fall 2016
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
More informationCourse 212: Academic Year Section 1: Metric Spaces
Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........
More informationMath 190: Fall 2014 Homework 4 Solutions Due 5:00pm on Friday 11/7/2014
Math 90: Fall 04 Homework 4 Solutions Due 5:00pm on Friday /7/04 Problem : Recall that S n denotes the n-dimensional unit sphere: S n = {(x 0, x,..., x n ) R n+ : x 0 + x + + x n = }. Let N S n denote
More informationAssignment #10 Morgan Schreffler 1 of 7
Assignment #10 Morgan Schreffler 1 of 7 Lee, Chapter 4 Exercise 10 Let S be the square I I with the order topology generated by the dictionary topology. (a) Show that S has the least uppper bound property.
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 information1 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 information(a) For an accumulation point a of S, the number l is the limit of f(x) as x approaches a, or lim x a f(x) = l, iff
Chapter 4: Functional Limits and Continuity Definition. Let S R and f : S R. (a) For an accumulation point a of S, the number l is the limit of f(x) as x approaches a, or lim x a f(x) = l, iff ε > 0, δ
More informationWeek 5 Lectures 13-15
Week 5 Lectures 13-15 Lecture 13 Definition 29 Let Y be a subset X. A subset A Y is open in Y if there exists an open set U in X such that A = U Y. It is not difficult to show that the collection of all
More informationIntroduction to Topology
Introduction to Topology Randall R. Holmes Auburn University Typeset by AMS-TEX Chapter 1. Metric Spaces 1. Definition and Examples. As the course progresses we will need to review some basic notions about
More informationREAL VARIABLES: PROBLEM SET 1. = x limsup E k
REAL VARIABLES: PROBLEM SET 1 BEN ELDER 1. Problem 1.1a First let s prove that limsup E k consists of those points which belong to infinitely many E k. From equation 1.1: limsup E k = E k For limsup E
More informationMA651 Topology. Lecture 10. Metric Spaces.
MA65 Topology. Lecture 0. Metric Spaces. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Linear Algebra and Analysis by Marc Zamansky
More informationReal Analysis. Joe Patten August 12, 2018
Real Analysis Joe Patten August 12, 2018 1 Relations and Functions 1.1 Relations A (binary) relation, R, from set A to set B is a subset of A B. Since R is a subset of A B, it is a set of ordered pairs.
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 informationMATH 23b, SPRING 2005 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Midterm (part 1) Solutions March 21, 2005
MATH 23b, SPRING 2005 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Midterm (part 1) Solutions March 21, 2005 1. True or False (22 points, 2 each) T or F Every set in R n is either open or closed
More informationSOLUTIONS TO THE FINAL EXAM
SOLUTIONS TO THE FINAL EXAM Short questions 1 point each) Give a brief definition for each of the following six concepts: 1) normal for topological spaces) 2) path connected 3) homeomorphism 4) covering
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 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 informationTOPOLOGY TAKE-HOME CLAY SHONKWILER
TOPOLOGY TAKE-HOME CLAY SHONKWILER 1. The Discrete Topology Let Y = {0, 1} have the discrete topology. Show that for any topological space X the following are equivalent. (a) X has the discrete topology.
More informationTopology, Math 581, Fall 2017 last updated: November 24, Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski
Topology, Math 581, Fall 2017 last updated: November 24, 2017 1 Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski Class of August 17: Course and syllabus overview. Topology
More informationProblems - Section 17-2, 4, 6c, 9, 10, 13, 14; Section 18-1, 3, 4, 6, 8, 10; Section 19-1, 3, 5, 7, 8, 9;
Math 553 - Topology Todd Riggs Assignment 2 Sept 17, 2014 Problems - Section 17-2, 4, 6c, 9, 10, 13, 14; Section 18-1, 3, 4, 6, 8, 10; Section 19-1, 3, 5, 7, 8, 9; 17.2) Show that if A is closed in Y and
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 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 General Topology Fall 2012 Homework 6 Solutions
Math 535 - General Topology Fall 202 Homework 6 Solutions Problem. Let F be the field R or C of real or complex numbers. Let n and denote by F[x, x 2,..., x n ] the set of all polynomials in n variables
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 informationMath 730 Homework 6. Austin Mohr. October 14, 2009
Math 730 Homework 6 Austin Mohr October 14, 2009 1 Problem 3A2 Proposition 1.1. If A X, then the family τ of all subsets of X which contain A, together with the empty set φ, is a topology on X. Proof.
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 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 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 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 informationSpring -07 TOPOLOGY III. Conventions
Spring -07 TOPOLOGY III Conventions In the following, a space means a topological space (unless specified otherwise). We usually denote a space by a symbol like X instead of writing, say, (X, τ), and we
More informationMarch 25, 2010 CHAPTER 2: LIMITS AND CONTINUITY OF FUNCTIONS IN EUCLIDEAN SPACE
March 25, 2010 CHAPTER 2: LIMIT AND CONTINUITY OF FUNCTION IN EUCLIDEAN PACE 1. calar product in R n Definition 1.1. Given x = (x 1,..., x n ), y = (y 1,..., y n ) R n,we define their scalar product as
More informationTopology Final Exam. Instructor: W. D. Gillam Date: January 15, 2014
Topology Final Exam Instructor: W. D. Gillam Date: January 15, 2014 Instructions: Print your name and Topology Final Exam in the upper right corner of the first page. Also be sure to put your name in 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 informationCHAPTER 7. Connectedness
CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set
More informationAfter taking the square and expanding, we get x + y 2 = (x + y) (x + y) = x 2 + 2x y + y 2, inequality in analysis, we obtain.
Lecture 1: August 25 Introduction. Topology grew out of certain questions in geometry and analysis about 100 years ago. As Wikipedia puts it, the motivating insight behind topology is that some geometric
More informationImmerse Metric Space Homework
Immerse Metric Space Homework (Exercises -2). In R n, define d(x, y) = x y +... + x n y n. Show that d is a metric that induces the usual topology. Sketch the basis elements when n = 2. Solution: Steps
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 information3 Hausdorff and Connected Spaces
3 Hausdorff and Connected Spaces In this chapter we address the question of when two spaces are homeomorphic. This is done by examining two properties that are shared by any pair of homeomorphic spaces.
More informationMath 361: Homework 1 Solutions
January 3, 4 Math 36: Homework Solutions. We say that two norms and on a vector space V are equivalent or comparable if the topology they define on V are the same, i.e., for any sequence of vectors {x
More information1 Selected Homework Solutions
Selected Homework Solutions Mathematics 4600 A. Bathi Kasturiarachi September 2006. Selected Solutions to HW # HW #: (.) 5, 7, 8, 0; (.2):, 2 ; (.4): ; (.5): 3 (.): #0 For each of the following subsets
More informationMetric Space Topology (Spring 2016) Selected Homework Solutions. HW1 Q1.2. Suppose that d is a metric on a set X. Prove that the inequality d(x, y)
Metric Space Topology (Spring 2016) Selected Homework Solutions HW1 Q1.2. Suppose that d is a metric on a set X. Prove that the inequality d(x, y) d(z, w) d(x, z) + d(y, w) holds for all w, x, y, z X.
More informationDifferential Topology Final Exam With Solutions
Differential Topology Final Exam With Solutions Instructor: W. D. Gillam Date: Friday, May 20, 2016, 13:00 (1) Let X be a subset of R n, Y a subset of R m. Give the definitions of... (a) smooth function
More informationINVERSE FUNCTION THEOREM and SURFACES IN R n
INVERSE FUNCTION THEOREM and SURFACES IN R n Let f C k (U; R n ), with U R n open. Assume df(a) GL(R n ), where a U. The Inverse Function Theorem says there is an open neighborhood V U of a in R n so that
More informationMath 541 Fall 2008 Connectivity Transition from Math 453/503 to Math 541 Ross E. Staffeldt-August 2008
Math 541 Fall 2008 Connectivity Transition from Math 453/503 to Math 541 Ross E. Staffeldt-August 2008 Closed sets We have been operating at a fundamental level at which a topological space is a set together
More informationContents. Index... 15
Contents Filter Bases and Nets................................................................................ 5 Filter Bases and Ultrafilters: A Brief Overview.........................................................
More informationSet, functions and Euclidean space. Seungjin Han
Set, functions and Euclidean space Seungjin Han September, 2018 1 Some Basics LOGIC A is necessary for B : If B holds, then A holds. B A A B is the contraposition of B A. A is sufficient for B: If A holds,
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 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 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 informationCOUNTABLE PRODUCTS ELENA GUREVICH
COUNTABLE PRODUCTS ELENA GUREVICH Abstract. In this paper, we extend our study to countably infinite products of topological spaces.. The Cantor Set Let us constract a very curios (but usefull) set known
More information1. Bounded linear maps. A linear map T : E F of real Banach
DIFFERENTIABLE MAPS 1. Bounded linear maps. A linear map T : E F of real Banach spaces E, F is bounded if M > 0 so that for all v E: T v M v. If v r T v C for some positive constants r, C, then T is bounded:
More informationMT804 Analysis Homework II
MT804 Analysis Homework II Eudoxus October 6, 2008 p. 135 4.5.1, 4.5.2 p. 136 4.5.3 part a only) p. 140 4.6.1 Exercise 4.5.1 Use the Intermediate Value Theorem to prove that every polynomial of with real
More informationAN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES
AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES DUSTIN HEDMARK Abstract. A study of the conditions under which a topological space is metrizable, concluding with a proof of the Nagata Smirnov
More information2 Metric Spaces Definitions Exotic Examples... 3
Contents 1 Vector Spaces and Norms 1 2 Metric Spaces 2 2.1 Definitions.......................................... 2 2.2 Exotic Examples...................................... 3 3 Topologies 4 3.1 Open Sets..........................................
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 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 informationSets and Functions. MATH 464/506, Real Analysis. J. Robert Buchanan. Summer Department of Mathematics. J. Robert Buchanan Sets and Functions
Sets and Functions MATH 464/506, Real Analysis J. Robert Buchanan Department of Mathematics Summer 2007 Notation x A means that element x is a member of set A. x / A means that x is not a member of A.
More informationMATH 31BH Homework 1 Solutions
MATH 3BH Homework Solutions January 0, 04 Problem.5. (a) (x, y)-plane in R 3 is closed and not open. To see that this plane is not open, notice that any ball around the origin (0, 0, 0) will contain points
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 informationMATH 225 Summer 2005 Linear Algebra II Solutions to Assignment 1 Due: Wednesday July 13, 2005
MATH 225 Summer 25 Linear Algebra II Solutions to Assignment 1 Due: Wednesday July 13, 25 Department of Mathematical and Statistical Sciences University of Alberta Question 1. [p 224. #2] The set of all
More informationFunctions as Relations
Functions as Relations Definition Recall that if A and B are sets, then a relation from A to B is a subset of A B. A function from A to B is a relation f from A to B with the following properties (i) The
More informationINDECOMPOSABILITY IN INVERSE LIMITS WITH SET-VALUED FUNCTIONS
INDECOMPOSABILITY IN INVERSE LIMITS WITH SET-VALUED FUNCTIONS JAMES P. KELLY AND JONATHAN MEDDAUGH Abstract. In this paper, we develop a sufficient condition for the inverse limit of upper semi-continuous
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 informationDifferential Topology Solution Set #2
Differential Topology Solution Set #2 Select Solutions 1. Show that X compact implies that any smooth map f : X Y is proper. Recall that a space is called compact if, for every cover {U } by open sets
More informationFilters in Analysis and Topology
Filters in Analysis and Topology David MacIver July 1, 2004 Abstract The study of filters is a very natural way to talk about convergence in an arbitrary topological space, and carries over nicely into
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 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 informationAxioms of separation
Axioms of separation These notes discuss the same topic as Sections 31, 32, 33, 34, 35, and also 7, 10 of Munkres book. Some notions (hereditarily normal, perfectly normal, collectionwise normal, monotonically
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 informationIntroduction to Proofs
Real Analysis Preview May 2014 Properties of R n Recall Oftentimes in multivariable calculus, we looked at properties of vectors in R n. If we were given vectors x =< x 1, x 2,, x n > and y =< y1, y 2,,
More informationMetric Spaces Lecture 17
Metric Spaces Lecture 17 Homeomorphisms At the end of last lecture an example was given of a bijective continuous function f such that f 1 is not continuous. For another example, consider the sets T =
More informationMath 147, Homework 1 Solutions Due: April 10, 2012
1. For what values of a is the set: Math 147, Homework 1 Solutions Due: April 10, 2012 M a = { (x, y, z) : x 2 + y 2 z 2 = a } a smooth manifold? Give explicit parametrizations for open sets covering M
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 informationx 0 + f(x), exist as extended real numbers. Show that f is upper semicontinuous This shows ( ɛ, ɛ) B α. Thus
Homework 3 Solutions, Real Analysis I, Fall, 2010. (9) Let f : (, ) [, ] be a function whose restriction to (, 0) (0, ) is continuous. Assume the one-sided limits p = lim x 0 f(x), q = lim x 0 + f(x) exist
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 informationHomework 5. Solutions
Homework 5. Solutions 1. Let (X,T) be a topological space and let A,B be subsets of X. Show that the closure of their union is given by A B = A B. Since A B is a closed set that contains A B and A B is
More informationIntroduction to generalized topological spaces
@ Applied General Topology c Universidad Politécnica de Valencia Volume 12, no. 1, 2011 pp. 49-66 Introduction to generalized topological spaces Irina Zvina Abstract We introduce the notion of generalized
More informationSection 21. The Metric Topology (Continued)
21. The Metric Topology (cont.) 1 Section 21. The Metric Topology (Continued) Note. In this section we give a number of results for metric spaces which are familar from calculus and real analysis. We also
More informationGeometry 2: Manifolds and sheaves
Rules:Exam problems would be similar to ones marked with! sign. It is recommended to solve all unmarked and!-problems or to find the solution online. It s better to do it in order starting from the beginning,
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 informationSolutions to Problem Set 5 for , Fall 2007
Solutions to Problem Set 5 for 18.101, Fall 2007 1 Exercise 1 Solution For the counterexample, let us consider M = (0, + ) and let us take V = on M. x Let W be the vector field on M that is identically
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 informationSolutions to Tutorial 7 (Week 8)
The University of Sydney School of Mathematics and Statistics Solutions to Tutorial 7 (Week 8) MATH2962: Real and Complex Analysis (Advanced) Semester 1, 2017 Web Page: http://www.maths.usyd.edu.au/u/ug/im/math2962/
More informationE.7 Alaoglu s Theorem
E.7 Alaoglu s Theorem 359 E.7 Alaoglu s Theorem If X is a normed space, then the closed unit ball in X or X is compact if and only if X is finite-dimensional (Problem A.25). Even so, Alaoglu s Theorem
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 informationCW complexes. Soren Hansen. This note is meant to give a short introduction to CW complexes.
CW complexes Soren Hansen This note is meant to give a short introduction to CW complexes. 1. Notation and conventions In the following a space is a topological space and a map f : X Y between topological
More informationAsetC X is closed if X \ C is open. Example: (a, b) is open in the metric space E 1 (R with the usual Euclidean metric). Given x (a, b), a<x<b.
Economics 204 Lecture 4 Thursday, July 30, 2009 Revised 7/31/09, Revisions Indicated by ** and Sticky Notes Section 2.4, Open and Closed Sets Definition 1 Let (X, d) be a metric space. A set A X is open
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 informationMAS331: Metric Spaces Problems on Chapter 1
MAS331: Metric Spaces Problems on Chapter 1 1. In R 3, find d 1 ((3, 1, 4), (2, 7, 1)), d 2 ((3, 1, 4), (2, 7, 1)) and d ((3, 1, 4), (2, 7, 1)). 2. In R 4, show that d 1 ((4, 4, 4, 6), (0, 0, 0, 0)) =
More informationConsequences of Continuity
Consequences of Continuity James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 4, 2017 Outline 1 Domains of Continuous Functions 2 The
More informationMATH 215 Sets (S) Definition 1 A set is a collection of objects. The objects in a set X are called elements of X.
MATH 215 Sets (S) Definition 1 A set is a collection of objects. The objects in a set X are called elements of X. Notation 2 A set can be described using set-builder notation. That is, a set can be described
More information0 Sets and Induction. Sets
0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set
More informationGeneral Topology. Summer Term Michael Kunzinger
General Topology Summer Term 2016 Michael Kunzinger michael.kunzinger@univie.ac.at Universität Wien Fakultät für Mathematik Oskar-Morgenstern-Platz 1 A-1090 Wien Preface These are lecture notes for a
More informationMath 535: Topology Homework 1. Mueen Nawaz
Math 535: Topology Homework 1 Mueen Nawaz Mueen Nawaz Math 535 Topology Homework 1 Problem 1 Problem 1 Find all topologies on the set X = {0, 1, 2}. In the list below, a, b, c X and it is assumed 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 informationDIFFERENTIAL GEOMETRY 1 PROBLEM SET 1 SOLUTIONS
DIFFERENTIAL GEOMETRY PROBLEM SET SOLUTIONS Lee: -4,--5,-6,-7 Problem -4: If k is an integer between 0 and min m, n, show that the set of m n matrices whose rank is at least k is an open submanifold of
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 informationMATH 411 NOTES (UNDER CONSTRUCTION)
MATH 411 NOTES (NDE CONSTCTION 1. Notes on compact sets. This is similar to ideas you learned in Math 410, except open sets had not yet been defined. Definition 1.1. K n is compact if for every covering
More information