Elementary topology of En

Abstract

In this chapter we explore various ramifications of the topological notions introduced in Section 1.4. We begin with some basic properties of functions, limits, and continuity. Then we turn in Sections 2.3 and 2.4 to some properties of En that depend essentially on the least upper bound axiom for real numbers. Among these properties are the convergence of Cauchy sequences (Theorem 2.2) and the Bolzano-Weierstrass theorem. In Section 2.6 a very general concept is introduced, that of topological space. In particular, any set S ⊂ En becomes a topological space with the relative topology which S inherits from En. The concepts of connected set and compact set are introduced in Sections 2.7 and 2.8. It turns out that these properties are preserved by continuous functions (Theorems 2.8 and 2.10). At the end of the chapter, the concept of metric space is introduced, and some interesting special cases are considered. Uniform convergence of a sequence of functions is studied as convergence in a certain metric on a space of bounded functions.