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.