Chapter 6 Metric Spaces Introduction The most important class of topological spaces is the class of metric spaces. Metric spaces provide a rich source of examples in topology. But more than this, most of the applications of topology to analysis are via metric spaces. The notion of metric space was introduced in 1906 by Maurice Fr´echet and developed and named by Felix Hausdorff in 1914 (Hausdorff [101]). 6.1 Metric Spaces 6.1.1 Definition. Let X be a non-empty set and d a real-valued function defined on X × X such that for a, b ∈ X: (i) d(a, b) ≥ 0 and d(a, b) = 0 if and only if a = b; (ii) d(a, b) = d(b, a); and (iii) d(a, c) ≤ d(a, b) + d(b, c), [the triangle inequality] for all a, b and c in X. Then d is said to be a metric on X, (X, d) is called a metric space and d(a, b) is referred to as the distance between a and b. 107 108 CHAPTER 6. METRIC SPACES 6.1.2 Example. The function d : R × R → R given by d(a, b) = |a − b|, a, b ∈ R is a metric on the set R since (i) |a − b| ≥ 0, for all a and b in R, and |a − b| = 0 if and only if a = b, (ii) |a − b| = |b − a|, and (iii) |a − c| ≤ |a − b| + |b − c|. (Deduce this from |x + y| ≤ |x| + |y|.) We call d the euclidean metric on R. 6.1.3 Example. The function d : R2 × R2 → R given by 2 2 d(ha1, a2i, hb1, b2i) = (a1 − b1) + (a2 − b2) p is a metric on R2 called the euclidean metric on R2. ... .... ..... .. .. ... ... ........ ......... 1 2 ......... hb , b i ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ha1, a2i ... ....... ........................ ......... 6.1.4 Example. Let X be a non-empty set and d the function from X × X into R defined by 0, if a = b d(a, b) = (1, if a =6 b. Then d is a metric on X and is called the discrete metric. 6.1. METRIC SPACES 109 Many important examples of metric spaces are “function spaces”. For these the set X on which we put a metric is a set of functions. 6.1.5 Example. Let C[0, 1] denote the set of continuous functions from [0, 1] into R. A metric is defined on this set by 1 d(f, g) = |f(x) − g(x)| dx 0 Z where f and g are in C[0, 1]. A moment’s thought should tell you that d(f, g) is precisely the area of the region which lies between the graphs of the functions and the lines x = 0 and x = 1, as illustrated below. 110 CHAPTER 6. METRIC SPACES 6.1.6 Example. Again let C[0, 1] be the set of all continuous functions from [0, 1] into R. Another metric is defined on C[0, 1] as follows: d∗(f, g) = sup{|f(x) − g(x)| : x ∈ [0, 1]}. Clearly d∗(f, g) is just the largest vertical gap between the graphs of the functions f and g. 6.1.7 Example. We can define another metric on R2 by putting ∗ d (ha1, a2i, hb1, b2i) = max{|a1 − b1|, |a2 − b2|} where max{x, y} equals the larger of the two numbers x and y. 6.1.8 Example. Yet another metric on R2 is given by d1(ha1, a2i, hb1, b2i) = |a1 − b1| + |a2 − b2|. 6.1. METRIC SPACES 111 A rich source of examples of metric spaces is the family of normed vector spaces. 6.1.9 Example. Let V be a vector space over the field of real or complex numbers. A norm k k on V is a map : V → R such that for all a, b ∈ V and λ in the field (i) k a k ≥ 0 and k a k= 0 if and only if a = 0, (ii) k a + b k ≤ k a k + k b k, and (iii) k λa k= |λ| k a k. A normed vector space (V, k k) is a vector space V with a norm k k. Let (V, k k) be any normed vector space. Then there is a corresponding metric, d, on the set V given by d(a, b) =k a − b k, for a and b in V . It is easily checked that d is indeed a metric. So every normed vector space is also a metric space in a natural way. For example, R3 is a normed vector space if we put 2 2 2 k hx1, x2, x3i k= x1 + x2 + x3 , for x1, x2, and x3 in R. q So R3 becomes a metric space if we put d(ha1, b1, c1i, ha2, b2, c2i) = k (a1 − a2, b1 − b2, c1 − c2) k 2 2 2 = (a1 − a2) + (b1 − b2) + (c1 − c2) . p Indeed Rn, for any positive integer n, is a normed vector space if we put 2 2 2 k hx1, x2, . , xni k= x1 + x2 + ··· + xn . q So Rn becomes a metric space if we put d(ha1, a2, . , ani, hb1, b2, . , bni) = k ha1 − b1, a2 − b2, . , an − bni k 2 2 2 = (a1 − b1) + (a2 − b2) + ··· + (an − bn) . p 112 CHAPTER 6. METRIC SPACES In a normed vector space (N, k k) the open ball with centre a and radius r is defined to be the set Br(a) = {x : x ∈ N and k x − a k< r}. This suggests the following definition for metric spaces: 6.1.10 Definition. Let (X, d) be a metric space and r any positive real number. Then the open ball about a ∈ X of radius r is the set Br(a) = {x : x ∈ X and d(a, x) < r}. 6.1.11 Example. In R with the euclidean metric Br(a) is the open interval (a − r, a + r). 2 6.1.12 Example. In R with the euclidean metric, Br(a) is the open disc with centre a and radius r. 6.1. METRIC SPACES 113 6.1.13 Example. In R2 with the metric d∗ given by ∗ d (ha1, a2i, hb1, b2i) = max{|a1 − b1|, |a2 − b2|}, the open ball B1(h0, 0i) looks like 2 6.1.14 Example. In R with the metric d1 given by d1(ha1, a2i, hb1, b2i) = |a1 − b1| + |a2 − b2|, the open ball B1(h0, 0i) looks like 114 CHAPTER 6. METRIC SPACES The proof of the following Lemma is quite easy (especially if you draw a diagram) and so is left for you to supply. 6.1.15 Lemma. Let (X, d) be a metric space and a and b points of X. Further, let δ1 and δ2 be positive real numbers. If c ∈ Bδ1 (a) ∩ Bδ2 (b), then there exists a δ > 0 such that Bδ(c) ⊆ Bδ1 (a) ∩ Bδ2 (b). The next Corollary follows in a now routine way from Lemma 6.1.15. 6.1.16 Corollary. Let (X, d) be a metric space and B1 and B2 open balls in (X, d). Then B1 ∩ B2 is a union of open balls in (X, d). Finally we are able to link metric spaces with topological spaces. 6.1.17 Proposition. Let (X, d) be a metric space. Then the collection of open balls in (X, d) is a basis for a topology τ on X. [The topology τ is referred to as the topology induced by the metric d, and (X, τ ) is called the induced topological space or the corresponding topological space or the associated topological space.] Proof. This follows from Proposition 2.2.8 and Corollary 6.1.16. 6.1.18 Example. If d is the euclidean metric on R then a basis for the topology τ induced by the metric d is the set of all open balls. But Bδ(a) = (a − δ , a + δ). From this it is readily seen that τ is the euclidean topology on R. So the euclidean metric on R induces the euclidean topology on R. 6.1. METRIC SPACES 115 6.1.19 Example. From Exercises 2.3 #1 (ii) and Example 6.1.12, it follows that the euclidean metric on the set R2 induces the euclidean topology on R2. 6.1.20 Example. From Exercises 2.3 #1 (i) and Example 6.1.13 it follows that the metric d∗ also induces the euclidean topology on the set R2. It is left as an exercise for you to prove that the metric d1 of Example 6.1.14 also induces the euclidean topology on R2. 6.1.21 Example. If d is the discrete metric on a set X then for each x ∈ X, B 1 (x) = 2 {x}. So all the singleton sets are open in the topology τ induced on X by d. Consequently, τ is the discrete topology. We saw in Examples 6.1.19, 6.1.20, and 6.1.14 three different metrics on the same set which induce the same topology. 6.1.22 Definition. Metrics on a set X are said to be equivalent if they induce the same topology on X. ∗ 2 So the metrics d, d , and d1, of Examples 6.1.3, 6.1.13, and 6.1.14 on R are equivalent. 6.1.23 Proposition. Let (X, d) be a metric space and τ the topology induced on X by the metric d. Then a subset U of X is open in (X, τ ) if and only if for each a ∈ U there exists an ε > 0 such that the open ball Bε(a) ⊆ U.