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Metric and Topological Spaces METRIC AND TOPOLOGICAL SPACES Part IB of the Mathematical Tripos of Cambridge This course, consisting of 12 hours of lectures, was given by Prof. Pelham Wilson in Easter Term 2012. This LATEX version of the notes was prepared by Henry Mak, last revised in June 2012, and is available online at http://people.pwf.cam.ac.uk/hwhm3/. Comments and corrections to [email protected]. No part of this document may be used for profit. Course schedules Metrics: Definition and examples. Limits and continuity. Open sets and neighbourhoods. Characterizing limits and continuity using neighbourhoods and open sets. [3] Topology: Definition of a topology. Metric topologies. Further examples. Neighbourhoods, closed sets, convergence and continuity. Hausdorff spaces. Homeomorphisms. Topological and non-topological properties. Completeness. Subspace, quotient and product topologies. [3] Connectedness: Definition using open sets and integer-valued functions. Examples, in- cluding intervals. Components. The continuous image of a connected space is connected. Path-connectedness. Path-connected spaces are connected but not conversely. Connected open sets in Euclidean space are path-connected. [3] Compactness: Definition using open covers. Examples: finite sets and Œ0; 1. Closed subsets of compact spaces are compact. Compact subsets of a Hausdorff space must be closed. The compact subsets of the real line. Continuous images of compact sets are compact. Quotient spaces. Continuous real-valued functions on a compact space are bounded and attain their bounds. The product of two compact spaces is compact. The compact subsets of Euclidean space. Sequential compactness. [3] 1 M ETRICSPACES Contents 2 1 Metric spaces 1.1 Introduction 5 1.2 Open balls and open sets 7 1.3 Limits and continuity 9 1.4 Completeness 9 2 Topological spaces 2.1 Introduction 13 2.2 Interiors and closures 2.3 Base of open subsets for a topology 14 2.4 Subspaces, quotients and products 16 3 Connectedness 3.1 Various results 19 3.2 Path-connectedness 20 3.3 Products of connected spaces 21 4 Compactness 4.1 Various results 25 4.2 Sequential compactness §1M ETRIC SPACES 1.1 Introduction Consider the Euclidean space Rn equipped with the standard Euclidean inner product: Given n Pn x; y R with coordinates xi ; yi respectively, we define .x; y/ i 1 xi yi , sometimes denoted by the2 dot product x y. ´ D 1 From this we have the Euclidean norm on Rn, x .x; x/ =2, representing the length of 1 k k ´ P 2 =2 the vector x. We have a distance function d2.x; y/ x y i .xi yi / , very often written as d simply. This is an example of a metric.´ k k D Definition 1.1. A metric space .X; d/ consists of a set X and a function, called the metric, d X X R such that, for all P; Q; R X: W ! 2 (i) d.P; Q/ 0, with equality iff P Q; > D (ii) d.P; Q/ d.Q; P /; D (iii) d.P; Q/ d.Q; R/ d.P; R/. C > Condition (iii) is called the triangle inequality. With the Euclidean metric, for any (possibly degenerate) triangle with vertices P; Q; R, the sum of the lengths of two sides of the triangle is at least the length of the third side. 2 1 . 1 I NTRODUCTION n Proposition 1.2. The Euclidean distance function d2 on R is a metric in the sense of Definition 1.1. Proof. (i) and (ii) are immediate. For (iii), use Cauchy-Schwarz inequality which says n !2 n ! n ! X X 2 X 2 xi yi 6 xi yi ; i 1 i 1 i 1 D D D or in inner product notation, .x; y/2 x 2 y 2 for x; y Rn. (See Lemma 1.3) 6 k k k k 2 R y x y C Q x P We take P to be the origin, Q with position vector x with respect to P , and R with position vector y with respect to Q. Then R has position vector x y with respect to P . Now C x y 2 x 2 2.x; y/ y 2 k C k D k k C C k k x 2 2 x y y 2 . x y /2: 6 k k C k k k k C k k D k k C k k This implies d.P; R/ x y x y d.P; Q/ d.Q; R/. D k C k 6 k k C k k D C Lemma 1.3. (Cauchy-Schwarz). .x; y/2 x 2 y 2 for any x; y Rn. 6 k k k k 2 Proof. For x 0, the quadratic polynomial in the real variable ¤ x y 2 2 x 2 2.x; y/ y 2 k C k D k k C C k k is non-negative for all . Considering the discriminant, we have 4.x; y/2 4 x 2 y 2. 6 k k k k Remarks. 1. In the Euclidean case, equality in the triangle inequality Q lies on the straight line segment PR. (See Example Sheet 1, Question 2) , 2. The argument for Cauchy-Schwarz above generalises to integrals. For example, if f; g are continuous functions on Œ0; 1, consider Z 1 ÂZ 1 Ã2 Z 1 Z 1 2 2 2 .f g/ > 0 fg 6 f g : 0 C ) 0 0 0 3 1 . 1 I NTRODUCTION More examples of metric spaces n Pn (i) Let X R , and d1.x; y/ i 1 xi yi or d .x; y/ maxi xi yi . These are both´ metrics. ´ D j j 1 ´ j j (ii) Let X be any set, and for x; y X, define the discrete metric to be 2 ( 1 if x y; ddisc.x; y/ ¤ ´ 0 if x y: D (iii) Let X C Œ0; 1 f Œ0; 1 R where f is continuous . We can define ´ ´ f W ! g metrics d1, d2, d on X by 1 Z 1 d1.f; g/ f g ; ´ 0 j j 1 ÂZ 1 à 2 2 d2.f; g/ .f g/ ; ´ 0 d .f; g/ sup f .x/ g.x/ : 1 ´ x Œ0;1 j j 2 For d2, the triangle inequality follows from Cauchy-Schwarz for integrals, i.e. R 2 R 2 R 2 fg 6 f g . See Remark 2 after Lemma 1.3, and use the same argument as in Proposition 1.2. (iv) British rail metric. Consider Rn with the Euclidean metric d, and let O denote the origin. Define a new metric on Rn by ( d.P; O/ d.O; Q/ if P Q; .P; Q/ C ¤ ´ 0 if P Q; D i.e. all journeys from P to Q P go via O. (All rail journeys go via London.) ¤ Some metrics in fact satisfy a stronger triangle inequality. A metric space .X; d/ is called ultra-metric if d satisfies condition (iii)0: d.P; R/ max d.P; Q/; d.Q; R/ 6 f g for all P; Q; R X. 2 Example. Let X Z and p be a prime. The p-adic metric is defined by ´ ( 0 if m n; dp.m; n/ D ´ 1=pr if m n where r max s N ps .m n/ : ¤ D f 2 W j g r1 We claim that d is an ultra metric. Indeed, suppose that dp.m; n/ 1=p and r2 r1 r2 D dp.n; q/ 1=p for distinct m; n; q Z. Then p .m n/ and p .n q/ together minD r1;r2 2 j j imply p .m q/. So, for some r min r1; r2 , f g j > f g r min r1;r2 dp.m; q/ 1=p 1=p f g D 6 max 1=pr1 ; 1=pr2 D f g max dp.m; n/; dp.n; q/ : D f g 4 1 . 2 O PENBALLSANDOPENSETS This extends to a p-adic metric on Q: For any rational x y, we can write x y r ¤ r D p m=n, where r Z, m; n are coprime to p, and define dp.x; y/ 1=p similarly. Then 2 D we also have .Q; dp/ as an ultra-metric space. n 1 Example. The sequence an 1 p p , where p is prime, is convergent in ´ 1C C C n .Q; dp/ with limit a .1 p/ . This is because dp.an; a/ 1=p for all n, and so ´ D dp.an; a/ 0 as n . ! ! 1 Lipschitz equivalence Definition 1.4. Two metrics 1; 2 on a set X are Lipschitz equivalent if 0 < 1 2 R such that 11 2 21. 9 6 2 6 6 n Remark. For metrics d1; d2; d on R , one can show that d1 > d2 > d > d2=pn > 1 1 d1=n, so they are all Lipschitz equivalent. (See Example Sheet 1, Question 9) Proposition 1.5. d1; d on C Œ0; 1 are not Lipschitz equivalent. 1 Proof. For n > 1, let fn C Œ0; 1 be as follows: 2 pn fn 0 1 2 1 n n d1.fn; 0/ Area of the triangle 1=pn 0 as n , while d .fn; 0/ pn as n . D D ! ! 1 1 D ! 1 ! 1 q 2 Exercise. Show that d2.fn; 0/ for all n, and so d2 is not Lipschitz equivalent to D 3 either d1 or d on C Œ0; 1. 1 1.2 Open balls and open sets Let .X; d/ be a metric space, and let P X, ı > 0.
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