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Metric Spaces and Continuity This Publication Forms Part of an Open University Module M303 Further pure mathematics Metric spaces and continuity This publication forms part of an Open University module. Details of this and other Open University modules can be obtained from the Student Registration and Enquiry Service, The Open University, PO Box 197, Milton Keynes MK7 6BJ, United Kingdom (tel. +44 (0)845 300 6090; email [email protected]). Alternatively, you may visit the Open University website at www.open.ac.uk where you can learn more about the wide range of modules and packs offered at all levels by The Open University. Note to reader Mathematical/statistical content at the Open University is usually provided to students in printed books, with PDFs of the same online. This format ensures that mathematical notation is presented accurately and clearly. The PDF of this extract thus shows the content exactly as it would be seen by an Open University student. 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ISBN 978 1 4730 2036 8 2.1 Contents Contents 1 Introducing metric spaces 5 1.1 The definition of a metric space 5 1.2 Examples of metric spaces 7 1.3 Understanding the geometry of metric spaces 12 2 Sequences in metric spaces 16 3 The definition of continuity in metric spaces 19 Solutions and comments on exercises 23 Index 27 1 Introducing metric spaces 1 Introducing metric spaces We now introduce the idea of a metric space, and show how this concept allows us to generalise the notion of continuity. We will then concentrate on looking at some examples of metric spaces and defer further discussion of continuous functions between metric spaces to the next chapter. You should not expect to have a firm grasp of the idea of a metric Subsection 1.1 shows how properties of the Euclidean distance function on space by the end of this section: n R can be generalised to define a metric on any set to create a metric this will come as you see more space. examples of metric spaces in Chapter 15 and later chapters. 1.1 The definition of a metric space Looking back at the definition of continuity for functions from Rn to Rm we see that it depends on: the Euclidean distance in Rn, d(n) Recall that if a, b Rk, then • (k) ∈ the Euclidean distance in Rm, d(m) d (a, b) is given by k • 2 n m i (bi ai) . the definition of convergence for sequences in both R and R . =1 − • qP Moreover, the definition of convergence for sequences also depends on the notion of distance. Thus a notion of distance (in both the domain and the codomain) is a fundamental part of our definition of continuity. Hence to define a notion of continuity for functions between arbitrary sets X and Y , we must first have effective notions of distance appropriate to the sets X and Y . An obvious strategy to do this is to try to find some properties of the Euclidean distance that we would expect any sensible notion of distance to possess. We showed that the Euclidean distance function has three properties. The Euclidean distance function d(n) : Rn Rn R has the following properties. × → For each a, b, c Rn: ∈ (M1) d(n)(a, b) 0, with equality holding if, and only if, a = b ≥ (M2) d(n)(a, b)= d(n)(b, a) (M3) d(n)(a, c) d(n)(a, b)+ d(n)(b, c) (the Triangle Inequality). ≤ Properties (M1)–(M3) do not use any special properties of the set Rn, so it seems reasonable to say that for any set X, a function d: X X R is a distance function if it satisfies them. It turns out that these× are indeed→ the appropriate properties of distance on which to base our definition. We call such a distance function a metric, and we call a set with such a function defined on it a metric space. The word metric comes from the Greek word µετρoν´ (metron), meaning ‘measure’, an instrument for measuring. 5 Metric spaces and continuity 1 Definition 1.1 Metric Let X be a set. A metric on X is a function d: X X R that satisfies the following three conditions. × → For each a, b, c X: ∈ (M1) d(a, b) 0, with equality holding if, and only if, a = b (M2) d(a, b) =≥ d(b, a) (M3) d(a, c) d(a, b) + d(b, c) (the Triangle Inequality). ≤ The set X, together with a metric d on X, is called a metric space, This definition was proposed by and is denoted by (X, d). Maurice Fr´echet, whose work is discussed in the History Reader. Remarks A ‘point’ may be nothing like a 1. Mathematicians usually refer to the members of the set X as points, dot in the plane: X could be a to emphasise the analogy with the points of a line, a plane or set of functions and then a point three-dimensional space. Similarly, mathematicians usually refer to in X would be a function. d(a, b) as the distance between a and b. The theory of metric spaces is the study of those properties of sets of points that depend only on distance. 2. Condition (M1) says that distance is a non-negative quantity, and that the only point of a metric space that is at zero distance from a given point is that point itself. 3. Condition (M2) says that the distance from a point a to a point b is precisely the same as the distance from b to a. (The metric is symmetric. It doesn’t matter whether you are going from a to b or b d(b, c) from b to a – the distance is the same.) 4. Condition (M3) tells us that d(a, c) gives the ‘shortest’ distance d(a, b) c between a and c. For if we go directly from a to c, that gives a distance of d(a, c). If we make a detour to b along the way, we must go d(a, c) d a, b b d b, c a a distance ( ) to get to , and then an additional distance ( ) to get from b to c (Figure 1.1). Condition (M3) tells us that the total Figure 1.1 Shortest distance distance travelled must then be at least as great as the ‘direct’ between a and c distance d(a, c). 5. Since our general definition is modelled on the corresponding properties of the Euclidean distance function d(n), it follows that (Rn, d(n)) is a metric space, for each n N.It is known as Euclidean n-space. Furthermore, in the context∈ of metric spaces, the Euclidean distance function d(n) is often referred to as the Euclidean metric for Rn. These are our first examples of metric spaces. If we look back at the proof of the Reverse Triangle Inequality for the Euclidean metric on Rn (Proposition ??), we see that the proof makes use only of properties (M1)–(M3) of the Euclidean metric. So it is no surprise to learn that there is a Reverse Triangle Inequality for any metric space. 6 1 Introducing metric spaces Proposition 1.2 Reverse Triangle Inequality for metric spaces Let (X, d) be a metric space. For each a, b, c X, ∈ (M3a) d(b, c) d(a, c) d(a, b) . The proof is essentially the same ≥ | − | as that for Proposition ??, and so we omit it. 1.2 Examples of metric spaces We now give a few examples of metric spaces with the aim of getting you used to showing that a distance function is a metric.
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