Metric Spaces

Abstract

Our goal in this chapter is to show that most of the concepts introduced in the previous chapters for the set ℝ of real numbers can be extended to any abstract metric space; i.e., a set on which the concept of metric (or distance) can be defined. Indeed, as we have already seen, the basic concept of limit which we studied in Chapters 2 and 3, and used to define (in Chapter 4) the related concept of continuity, is defined in terms of distance. Let us recall that the distance between two real numbers x and y is defined to be d(x, y) := |x - y|, and satisfies three simple properties: For any numbers x, y, z ∈ ℝ we have: (i) |x - y| ≥ 0, and equality holds if and only if x = y; (ii) |x - y| = |y − x|; and (iii) |x − y| ≤ |x − z| + |y − z|. Property (iii) is called the Triangle Inequality for obvious geometrical reasons. Using this distance, we defined, in Chapter 2, the concepts of ε-neighborhood, open set, closed set, limit point, isolated point, convergent sequence and Cauchy sequence. We then defined the concept of limit for general real-valued functions of a real variable, and proved that such limits can also be defined in terms of limits of sequences. Also, before introducing the related notion of continuity, we introduced (in Chapter 4) the topological concepts of compactness and connectedness. All these notions can be defined, in essentially the same way, in any (abstract) metric space and the proofs of most theorems are basically copies of the ones we gave for the special metric space ℝ if one replaces |x − y| by d(x, y) throughout. Many proofs will therefore be brief or will be left as exercises for the reader.