Topology of the Real Line

Abstract

In the field of Analysis the concepts of the limit and the continuity of a function f at a point x = a are defined in terms of open intervals. For example, the condition | f(x) − L | < e says that f(x) is in an open interval centered at L, and the condition | x − a | < δ says that x is in an open interval centered at a. These intervals are specified in terms of the distance between x and y given by | x − y | . Topology is a branch of Mathematics where these concepts are extended to spaces where one can discuss intervals without having to rely on a distance formula. As a result the concepts of limit and continuity can be extended to such spaces, and it can be shown that many of the properties associated with continuous functions defined on the real line are shared by continuous functions defined on these more general spaces. Although the theorems discussed in this chapter are presented in the context of sets on the real line, virtually all of the theorems are true in the more general context of any topological space. Many of the techniques used to prove these theorems are the same techniques one would use for a general topological space, and, therefore, this chapter can be thought of as an introduction to the field of Topology even though general topological spaces are not discussed here.