Metric Spaces

Abstract

The topological spaces introduced in Chapter 2 allowed us to discuss the notion of proximity and introduce basic concepts of modern analysis, such as compactness, connectedness, and continuity. We have studied extensively one specific example, the set of real numbers. In this chapter, we extend the discussion of these topics by introducing the concept of metric space. A metric space is essentially a set of elements, or points, together with a real-valued nonnegative function, which measures the distance between them. The metric structure brings our discussion closer to the typical spaces of mathematical calculus. With the notion of distance, the formalisation of the idea of proximity goes beyond the qualitative approach of topology, so to speak, to take a quantitative flavour. It turns out that the metric spaces are Hausdorff first-countable topological spaces. Thus, many results that we have derived in Chapter 2 can be fruitfully applied to their study. However, some key aspects of metric spaces are investigated using the notion of sequence, introduced in Chapter 5, so they will be postponed to Section 5.2.