ABSTRACT
A bounded interval is closed “if it contains both endpoints.” There is a more general notion of a closed set, which is introduced in this chapter. The chapter explores properties of the underlying space R, which are usually referred to as topological properties of R. It proves two theorems that address the unions and intersections of closed sets: the union of a finite number of closed sets is a closed set. The chapter provides a discussion of the topology of R with the idea of an accumulation point of a set. In a certain sense, an accumulation point of a set is a point that is “very close” to infinitely many elements of S, and the chapter introduces a definition that captures this idea of closeness. It introduces the definition (after first defining neighborhoods in R), and proves several important theorems about accumulation points.
