Axioms for ℝ

Abstract

As you have seen in the previous chapter, continuity and convergence are basic for topology, and calculus is not sufficient as a background. The latter statement has a double meaning. We need more calculus-like theorems, especially on functions of several variables, but that is not all: Topology also requires a more precise kind of reasoning than an introductory Calculus course. We will have to prove all of our theory and we cannot afford to rely on pictures (although they will be an invaluable aid). In this chapter, we lay the foundations for a rigorous theory in the form of a system of axioms.