ABSTRACT
The notion of a compact set, which we introduce in this chapter, is one of the most important in mathematics. It is an important ingredient in the proof of some of the fundamental theorems of analysis, such as the theorem which states that a continuous real-valued function on a closed, bounded interval attains its maximum. In addition to compactness, the chapter is also concerned with several other topics including the notion of a convergent sequence in a normed vector space and least upper bounds and greatest lower bounds of subsets of R. These concepts are needed for the study of compact sets. We also define uniform continuity and connectedness. The notion of connectedness is central to the proof of another basic theorem, the Intermediate Value Theorem.
