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Real Analysis with Economic Applications - Chapter C

Chapter C Metric Spaces This chapter provides a self-contained review of the basic theory of metric spaces. Chances are good that you are familiar with the rudiments of this theory, so our exposition starts a bit faster than usual. But don’t worry, we slow down when we get to the “real stuff,” | Chapter C Metric Spaces This chapter provides a self-contained review of the basic theory of metric spaces. Chances are good that you are familiar with the rudiments of this theory so our exposition starts a bit faster than usual. But don t worry we slow down when we get to the real stuff that being the analysis of the properties of connectedness separability compactness and completeness for metric spaces. Connectedness is a geometric property that will be of limited use in this course. Consequently its discussion here is quite brief all we do is to identify the connected subsets of R and prepare for the Intermediate Value Theorem that will be given in the next chapter. Our treatment of separability is also relatively short even though this concept will be important for us later on. Because separability usually makes an appearance only in relatively advanced contexts we will study this property in greater detail later. Utility theory that we sketched out in Section B.4 can be taken to the next level with the help of even an elementary investigation of connected and separable metric spaces. As a brief application therefore we formulate here the metric versions of some of the utility representation theorems that were proved in that section. The story will be brought to its conclusion in Chapter D. The bulk of this chapter is devoted to the analysis of metric spaces that are either compact or complete. A good understanding of these two properties is essential for real analysis and optimization theory so we spend quite a bit of time studying them. In particular we consider several examples give two proofs of the Heine-Borel Theorem for good measure and discuss why closed and bounded spaces need not be compact in general. Totally bounded sets the sequential characterization of compactness and the relationship between compactness and completeness are also studied with care. Most of the results established in this chapter are relatively preliminary observations whose main