DOCSLIB.ORG

September 2020 INTRODUCTION to TOPOLOGY (SOME ADDITIONAL

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
September 2020 INTRODUCTION to TOPOLOGY (SOME ADDITIONAL September 2020 INTRODUCTION TO TOPOLOGY (SOME ADDITIONAL BASIC EXERCISES) MICHAEL MEGRELISHVILI Abstract. We provide some additional exercises in the course Topology-8822205 (Bar-Ilan University). We are going to update this file several times (during the current semester). Contents 1. Metric spaces 1 2. Topological spaces 5 3. Topological products 8 4. Compactness 9 5. Quotients 11 6. Hints and solutions 12 References 25 1. Metric spaces Exercise 1.1. Give an example of a metric space (X; d) containing two balls such that B(a1; r1) ( B(a2; r2) but r2 < r1: Hint: think about metric subspaces of R. Exercise 1.2. Let (X; d) be a metric space and 0 < r + d(a; b) < R: Prove that B(b; r) ⊆ B(a; R). Conclude that the ball B(a; R) is an open subset of X. Exercise 1.3. Let X be a set. Define ( 0 for x = y d (x; y) := ∆ 1 for x 6= y Show that (X; d∆) is an ultrametric space and describe the balls and spheres according to their radii. Date: September 14, 2020. 1 2 Exercise 1.4. Find non-isometric metric spaces X; Y such that there exist isometric embeddings f : X,! Y , g : Y,! X. Exercise 1.5. Let (V; jj · jj) be a normed space. Show that every translation Tz : V ! V; Tz(x) = z + x is an isometry. Conclude that all open balls B(a; r) with the same radius r (and a 2 V ) are isometric. Exercise 1.6. Give geometric descriptions of B[v; r];B(v; r);S(v; r) in R2 for v = (1; 2), r = 3 with respect to the following metrics: (a) Euclidean d; (b) d1; (c) dmax. Exercise 1.7. (1) Consider the normed spaces (C[0; 1]; jj · jjmax) and (C[0; 1]; jj · jj1). Give an intuitive description of the "closed ball" Bmax[θ; 3], B1[θ; 3] with radius r = 3 and the center in the zero function θ : [0; 1] ! R; x 7! 0. (2) Explain theoretically why B1[θ; 3] ⊂ Bmax[θ; 3]. Exercise 1.8. Let (X; d) be a pseudometric space. Prove that the following conditions are equivalent: (1)( X; d) is a metric space. (2) Every finite subset F ⊂ X is closed. (3) For every converging sequence f : ! X; xn := f(n) the limit lim xn is N n!1 unique. (4)( X; d) satisfies the Hausdorff property. (5) \fB(a; r)j r > 0g = fag. Exercise 1.9. (p-adic metric on Z) For every given prime p define on the set Z of all integers the following metric ( 0 for x = y dp(x; y) := 1 i pk for k = k(x; y) = maxfi : p j(x − y)g Show that (1) Is an ultrametric and diam(Z; dp) = 1. Hint: Observe that k(x; z) ≥ minfk(x; y); k(y; z)g: (2) Every translation Tz : Z ! Z;Tz(x) = z + x is an isometry. n (3) lim p = 0. Conclude: ( ; dp) does not contain isolated points. n!1 Z (4) Every ball B(0; r) (center is 0) is a subgroup of (Z; +). Every ball B(a; r) is clopen. (5) For every ball B(a; r) and every b 2 B(a; r) we have B(a; r) = B(b; r). That is, every point inside a ball is its center ! Try to generalize to every ultrametric space. Remark: see the schematic pictures of (Z; d3) and (Z; d7) Figures 1 and 2. (6) Describe the "next stage" in the picture for (Z; d3). 3 Figure 1. p=3 Figure 2. p=7 Exercise 1.10. (Cantor cube) Let N X = f0; 1g = fx = (x1; x2; ··· )j xk 2 f0; 1gg be the set of all binary sequences. Define d : X × X ! [0; 1) as ( 0 for x = y d(x; y) := 1 k for k = k(x; y) = minfi : xi 6= yig Show that (1) d is an ultrametric and diam(X; d) = 1. Hint: Observe that k(x; z) ≥ minfk(x; y); k(y; z)g (2) Give an example of a converging sequence with distinct members (hence, is not eventually constant). (3) Every ball B(θ; r) (center is the zero-sequence θ := (0; 0; ··· )) is a subgroup of the group (f0; 1gN; +) (where the sum + of sequences is "modulo 2"). Exercise 1.11. (Hilbert space l2) 4 Define 1 X 2 l2 := fx = (x1; x2; ··· )j xi 2 R xi < 1g i=0 v u 1 uX 2 jjxjj := t xi i=0 P1 Then l2 is a normed space and hx; yi := i=0 xiyi is an inner (scalar) product. Show that (1) There exists a sequence in l2 which has no limit but converges coordinate-wise (in contrast to the Euclidean space Rn). n (2) There exist isometric embeddings R ,! l2, (N; d∆) ,! l2. ( 0 for x = y Recall that d (x; y) := ∆ 1 for x 6= y (3) Find a linear isometric embedding f : l2 ! l2 which is not onto. (4) ∗ There exists a metric space M with 4 elements which cannot be embedded n isometrically into l2 (and hence into any R ). Show also that 4 is the minimum possible natural number with above property. Hint: when d(x; z) = d(x; y) + d(y; z) in R2, Rn ? Exercise 1.12. Let (M; d) be a metric space. Show that for every closed subset A ⊆ M there exists a sequence On of open subsets in M such that A = \nOn. Exercise 1.13. Show that the p-adic metric space (Z; dp) (see Exercise 1.9) is not complete. What about the Cantor cube (from Exercise 1.10) ? Exercise 1.14. Show that the normed space (C[−1; 1]; jj · jj1) defined by Z 1 jjfjj1 = jf(x)jdx −1 is not a Banach space. Exercise 1.15. Give an example of a linear map f : E1 ! E2 between two normed spaces which is not continuous. Exercise 1.16. (Cantor Set) Let C be the set of real numbers that are sums of series of the form 1 X ak where k 2 f0; 2g: 3k k=1 In other words, C consists of the real numbers that have the form 0:a1a2:::ak::: without the digit 1 in the number system with base 3. Prove that (1) C is contained in [0; 1]. 1 2 (2) C does not meet ( 3 ; 3 ). 3s+1 3s+2 (3) C does not meet ( 3k ; 3k ) for every s; k 2 N. (4) Find a geometric description of C (Hint: removing from [0; 1] countably many open intervals). Remark: See for example [1], [5] or/and wikipedia. 5 Totally bounded metric spaces. A metric space (M; d) is said to be totally bounded (or, precompact) if for every " > 0 there exists a finite subset A" of X such that A" is "-dense in (M; d). A subset Y of M is said to be totally bounded if the metric subspace (Y; dY ) is totally bounded. Exercise 1.17. Let (M; d) be a metric space. (1) Every finite subset is totally bounded; (2) Every totally bounded subset is bounded; (3) Finite union of totally bounded subsets is totally bounded; (4) If X is a totally bounded subset of M then every subset Y of X is also totally bounded. (5) The closure cl(Y ) of a totally bounded subset Y is also totally bounded; Exercise 1.18. (1) Let (X; d) be a complete metric space and Y ⊂ X is a closed subset. Then the metric subspace (Y; dY ) is also complete. (2) Let (Y; dY ) is a metric subspace of (X; d). Show that if (Y; dY ) is complete then Y is closed in X. 2. Topological spaces Exercise 2.1. Prove that for every pseudometric space (X; d) the pair (X; top(d)) is a topological space. Exercise 2.2. Prove that any intersection \iτi of topologies τi on the same set X is a topology. Show that it is not true, in general, for unions. Exercise 2.3. Let (X; τ) be a topological space. Show that the following conditions are equivalent: (1)( X; τ) 2 T1; (2) Every singleton fag is closed; (3) Every finite subset F ⊂ X is closed; (4) τcofin ⊂ τ. Exercise 2.4. On Z define the following family τ≤ of subsets τ≤ := f;; Z;Ok : k 2 Zg where Ok := fx 2 Z : x ≤ kg: Show that: (1)( Z; τ≤) is a connected topological space with property T0. (2) It is not (pseudo)metrizable. (3) Every continuous map (Z; τ≤) ! R into the reals is constant. Exercise 2.5. On Z describe the smallest topology τcofin with the following property: every singleton fag is closed in Z. Show that (Z; τcofin) is connected, not Hausdorff and not (pseudo)metrizable. Exercise 2.6. (generalized subspace topology) Let (X; τ) be a topological space and f : Y ! X be a function. Define −1 τf;Y := ff (O): O 2 τg: 6 Prove that (Y; τf;Y ) is a topological space. Remark: In the case of a subset Y ⊂ X and the inclusion f = in : Y ! X we get the subspace topology on Y . Exercise 2.7. (heredity of the continuity) Let f : X ! Y be a continuous map between topological spaces and X1 ⊂ X; Y1 ⊂ Y be subspaces such that f(X1) ⊂ Y1. Show the continuity of the following induced map f∗ : X1 ! Y1; x 7! f(x): Conclude that, in particular, the following induced maps are continuous: (1) Y ! Z; y 7! f(y) (2) Y ! f(Y ); y 7! f(y) are also continuous.
Load more

Recommended publications
  • CLASS NOTES MATH 551 (FALL 2014) 1. Wed, Aug. 27 Topology Is the Study of Shapes. (The Greek Meaning of the Word Is the Study Of
    CLASS NOTES MATH 551 (FALL 2014) BERTRAND GUILLOU 1. Wed, Aug. 27 Topology is the study of shapes. (The Greek meaning of the word is the study of places.) What kind of shapes? Many are familiar objects: a circle or triangle or square. From the point of view of topology, these are indistinguishable. Going up in dimension, we might want to study a sphere or box or a torus. Here, the sphere is topologically distinct from the torus. And neither of these is considered to be equivalent to the circle. One standard way to distinguish the circle from the sphere is to see what happens when you remove two points. One case gives you two disjoint intervals, whereas the other gives you an (open) cylinder. The two intervals are disconnected, whereas the cylinder is not. This then implies that the circle and the sphere cannot be identified as topological spaces. This will be a standard line of approach for distinguishing two spaces: find a topological property that one space has and the other does not. In fact, all of the above examples arise as metric spaces, but topology is quite a bit more general. For starters, a circle of radius 1 is the same as a circle of radius 123978632 from the eyes of topology. We will also see that there are many interesting spaces that can be obtained by modifying familiar metric spaces, but the resulting spaces cannot always be given a nice metric. ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? As we said, many examples that we care about are metric spaces, so we'll start by reviewing the theory of metric spaces.
    [Show full text]
  • Which Topologies Can Have Immediate Successors in the Lattice of T1-Topologies?
    Applied General Topology c Universidad Polit´ecnica de Valencia @ Volume 5, No. 2, 2004 pp. 231-242 Which topologies can have immediate successors in the lattice of T 1-topologies? Ofelia T. Alas and Richard G. Wilson∗ Abstract. We give a new characterization of those topologies which have an immediate successor or cover in the lattice of T1-topologies on a set and show that certain classes of compact and countably compact topologies do not have covers. Keywords: lattice of T1-topologies, maximal point, KC-space, cover of a topology, upper topology, lower topology 2000 AMS Classification: Primary 54A10; Secondary 06A06 1. Introduction and Preliminary Results The lattice L1(X) of T1-topologies on a set X has been studied, with differing emphasis, by many authors. Most articles on the subject have considered com- plementation in the lattice and properties of mutually complementary spaces (see for example, [12, 13, 15]. Here we consider a facet of the order structure of the lattice L1(X), specifically the problem of when a jump can occur in the order; that is to say, when there exist topologies τ and τ + on a set X such that whenever µ is a topology on X such that τ ⊆ µ ⊆ τ + then µ = τ or µ = τ +. The existence of jumps in L1(X) has been studied in [10] and [16] where the immediate successor τ + was said to be a cover of (or simply to cover) τ. We prefer order-theoretic terminology and then call τ a lower topology for X; the topology τ + will then be termed an upper topology.
    [Show full text]
  • Chapter 6 Products and Quotients
    Chapter VI Products and Quotients 1. Introduction In Chapter III we defined the product of two topological spaces \] and and developed some simple properties of the product. (See Examples III.5.10-5.12 and Exercise IIIE20. ) Using simple induction, that work generalized easy to finite products. It turns out that looking at infinite products leads to some very nice theorems. For example, infinite products will eventually help us decide which topological spaces are metrizable. 2. Infinite Products and the Product Topology The set \‚] was defined as ÖÐBßCÑÀB−\ßC −]×. How can we define an “infinite product” set ? Informally, we want to say something like \œ Ö\α Àα −E× \œ Ö\αααα Àα −EלÖÐBÑÀB −\ × so that a point BB\ in the product consists of “coordinates αα” chosen from the 's. But what exactly does a symbol like BœÐBÑα mean if there are “uncountably many coordinates?” We can get an idea by first thinking about a countable product. For sets \"# ß \ ß ÞÞÞß \ 8,... we can informally define the product set as a certain set of sequences: ∞ . \œ8œ" \8888 œÖÐBÑÀB −\ × But if we want to be careful about set theory, then a legal definition of \ should have the form \œÖÐBÑ−Y888 ÀB −\ ×. From what “pre-existing set” Y will the sequences in \ be chosen? The answer is easy: we write ∞∞ \œ8œ" \8888 œÖB−Ð 8œ" \Ñ ÀBÐ8ÑœB −\×Þ Thus the elements of ∞ are certain functions (sequences) defined on the index set . This idea 8œ"\8 generalizes naturally to any productÞ Definition 2.1 Let Ö\α Àα − E× be a collection of sets.
    [Show full text]
  • THREE DIMES of TOPOLOGY A. Candel Class Notes for Math 262, Winter 95-96, the University of Chicago. Preface Chapter I. Topologi
    THREE DIMES OF TOPOLOGY A. Candel Class Notes for Math 262, Winter 95-96, The University of Chicago. Preface Chapter I. Topological spaces 1. The concept of topological space 2. Bases and subbases 3. Subspaces, unions and hyperspaces 4. Pseudometric and metric spaces 5. The order topology 6. More examples Chapter II. Continuous maps 1. Continuous maps 2. Continuity and convergence Chapter III. Product and quotient spaces 1. The product topology 2. Quotient spaces 3. Cutting and pasting Chapter IV. Connected and path-connected spaces 1. Connected spaces 2. Path-connected spaces 3. Local connectedness and path-connectedness Chapter V. Convergence 1. Hausdor® spaces 2. Countability axioms 3. Filters 4. Ultra¯lters Chapter VI. Compact spaces 1. Compact spaces 2. The Tichonov theorem 3. Local compactness 4. Compacti¯cation 5. Components of a compact hausdor® space Chapter VII. The Urysohn lemma 1. Regular and normal spaces Typeset by AMS-TEX 1 2 TOPOLOGY 2. Urysohn and Tietze lemmas 3. Embeddings 4. Stone-Cec· h compacti¯cation 5. Metrization Chapter VIII. Paracompact spaces 1. Paracompact spaces 2. Partitions of unity Chapter IX. Function spaces 1. Topologies on function spaces 2. The evaluation map 3. Uniform spaces 4. Ascoli's theorem Chapter X. Topological groups 1. Topological groups 2. Topological vector spaces 3. Haar measure 4. Stone-Weierstrass theorem 5. Topological spaces and Banach algebras Homeworks and exams Bibliography NOTE: These notes are being o®ered without warranty. Bibliography and references have not been collected. Chapter X does not exist. 1. THE CONCEPT OF TOPOLOGICAL SPACE 1 CHAPTER I TOPOLOGICAL SPACES 1.
    [Show full text]
  • MA30055 Introduction to Topology, Spring 2017
    MA30055 Introduction to Topology, Spring 2017 G.K. Sankaran February 6, 2018 What this course is about This course is about topology, which is really not a subject you have yet seen. The prerequisites tell you that you need metric spaces: that's true (just about) but misleading, because it suggests that it is yet another analysis course. It isn't an analysis course. There is barely an " in sight and we shan't differentiate anything in earnest. The ideas and patterns of thinking often (not always) resemble things you have seen in algebra; but it's not an algebra course either. It is often said that a topologist is a person who cannot distinguish between a doughnut and a coffee cup, because they both have one hole. Then you are told that topology is about things that don't change when you stretch or bend spaces. That's true, at last partly, but it isn't the mechanism we use for thinking about topology. A lot of the time we use, quite simply, sets: topology is rather fundamental in the same way that set theory is. De Morgan's rules will turn up from time to time. The way we present topology in this course uses the language of categories, which is another way of looking at the fundamentals of mathematics. Part of the idea is to give a (very light) introduction to category theory, but in a context. Category theory is hugely general (almost everything is a category), but topology is a good place to encounter it, as well as being historically the place it came from.
    [Show full text]
  • Pseudo Perfectly Continuous Functions and Closedness/Compactness of Their Function Spaces
    Applied General Topology c Universidad Polit´ecnica de Valencia @ Volume 14, no. 1, 2013 pp. 115-134 Pseudo perfectly continuous functions and closedness/compactness of their function spaces J. K. Kohli, D. Singh, Jeetendra Aggarwal and Manoj Rana Abstract A new class of functions called ‘pseudo perfectly continuous’ functions is introduced. Their place in the hierarchy of variants of continuity which already exist in the literature is highlighted. The interplay between topological properties and pseudo perfect continuity is investigated. Function spaces of pseudo perfectly continuous functions are considered and sufficient conditions for their closedness and compactness in the topology of pointwise convergence are formulated. 2010 MSC: 54C05, 54C10, 54C35, 54G15, 54G20,54D10 Keywords: (quasi) perfectly continuous function, Dδ -supercontinuous func- tion, dδ-map, slightly continuous function, pseudo-partition topology, Dδ T0-space, δ-completely regular space, Alexandroff space (≡ saturated space). 1. Introduction The class of pseudo perfectly continuous functions properly contains the class of quasi perfectly continuous functions [36] which in its turn strictly contains the class of δ-perfectly continuous functions [27] and so includes all perfectly continuous functions due to Noiri [48] and hence contains all strongly continu- ous functions of Levine [37]. It is well known that the set C(X, Y ) of all continuous functions from a space X into a space Y is not closed in Y X in the topology of pointwise convergence. However, Naimpally [46] showed that in contrast to continuous functions the set S(X, Y ) of strongly continuous functions is closed in Y X in the topology 116 J.K.Kohli,D.Singh,J.AggarwalandM.Rana of pointwise convergence if X is locally connected and Y is Hausdorff.
    [Show full text]
  • 3 Countability and Connectedness Axioms
    3 COUNTABILITY AND CONNECTEDNESS AXIOMS Definition 3.1 Let X be a topological space. A subset D of X is dense in X iff D¯ = X. X is separable iff it contains a countable dense subset. X satisfies the first axiom of countability or is first countable iff for each x ∈ X, there is a countable neighbourhood basis at x. Any metrisable space is first countable. X satisfies the second axiom of countability or is second countable iff the topology of X has a countable basis. For example, R is separable, Q forming a countable dense subset. R is even second countable, for example the countable collection of open intervals with rational end-points forms a basis. Lemma 3.2 Let D be a subset of a space X and let B be a basis of open sets not containing ∅. Then D is dense in X iff for each B ∈ B, B ∩ D 6= ∅. Proof. ⇒: Suppose that B ∈ B but B ∩ D = ∅. Then D is contained in the closed set X − B, so D¯ 6= X. ⇐: Let x ∈ X and let N be a neighbourhood of x in X. Then there is B ∈ B so that x ∈ B ⊂ N. Since B ∩ D 6= ∅, it follows that N ∩ D 6= ∅, so by Proposition 1.17 x ∈ D¯, so that D¯ = X. Theorem 3.3 Let X be second countable. Then X is separable and first countable. Proof. Let B be a countable basis: we may assume that ∅ ∈/ B. For each B ∈ B, choose xB ∈ B. Then by Lemma 3.2, {xB /B ∈ B} is a countable dense subset of X, so X is separable.
    [Show full text]
  • Selected Solutions February 2019
    Math 590 Final Exam Practice Questions Selected Solutions February 2019 1. Each of the following statements is either true or false. If the statement holds in general, write \True". Otherwise, write \False". No justification necessary. (i) Let S be the set of sequence of rational numbers that are eventually zero. Then S is countable. True. Hint: Write S as a countable union of countable sets. (ii) Let S be the set of sequences of rational numbers that converge to zero. Then S is count- able. False. Hint: Apply Cantor's diagonalization argument to the subset of S of sequences n ( n )n2N for n 2 f0; 1g. (iii) Given a countable collection of uncountable sets A0 ⊇ A1 ⊇ A2 ⊇ · · · , their intersection must be uncountable. 1 1 False. Hint: Consider the sets (− n ; n ) ⊆ R. (iv) Let X be a space and A ⊆ X. If A is closed in X, then A will also be closed with respect to any finer topology on X. True. Hint: The complement of A will still be open in any finer topology. (v) Let X be a space and A ⊆ X. If A is closed in X, then A will also be closed with respect to any coarser topology on X. False. Hint: Consider [0; 1] ⊆ R with respect to the standard topology and the in- discrete topology. (vi) Let X be a set. If T1 is a finer topology on X than T2, then the identity map on X is necessarily continuous when viewed as function from (X; T1) ! (X; T2). True. Hint: Verify that the preimage of any open set is open.
    [Show full text]