The Real Line

Abstract

Abstract The real line is the playground of real analysis. Although many introductory treatments of the real line might lead one to believe that the real line is rather lackluster and dull, such is not the case. In this chapter we will explore several pathological properties and subsets of the real line. Before proceeding, however, we will first present some useful definitions. A subset P of JR is said to be perfect if it is closed and if every point of P is a limit point. A subset A of JR is said to be nowhere dense if its closure has an empty interior. A subset of JR is said to be of the first category if it is a countable union of nowhere dense sets and otherwise is said to be of the second category. A subset of JR is said have the property of Baire if it is given by the symmetric difference of an open set and a set of the first category. A Hamel basis H for JR over Q is a subset of JR such that each nonzero real number may be uniquely represented as a finite linear combination of distinct elements from H with nonzero rational coefficients. That such a set exists is a consequence of the axiom of choice and is shown in Example 1.29.