Arnoud van Rooij

Open and Closed Sets

Proof: (O1) ∅ is open because the condition (1) is vacuously satisfied: there is no x ∈ ∅. X is open because any ball is by definition a subset of X. (O2) Let Si be an open set for i = 1, . . . , n, and let x ∈ ∩i=1Si. We must find an > 0 for which B(x, ) ⊆ ∩i=1Si. For each i there is an i such that B(x, i) ⊆ Si, because each Si is open. Let = min{ 1, . . . , n}. Then 0 < 5 i for each i; therefore B(x, ) ⊆ B(x, i) ⊆ Si for each i — i.e., B(x, ) ⊆ ∩i=1Si. (O3) Let x ∈ ∪α∈ASα; we must find an > 0 for which B(x, ) ⊆ ∪α∈ASα. Since x ∈ ∪α∈ASα, we have x ∈ Sᾱ for some ᾱ. Since Sᾱ is open, there is an > 0 such that B(x, ) ⊆ Sᾱ ⊆ ∪α∈ASα. ‖

Transition to Topology

We start working on the program that was indicated in the preceding chapter: We wish to reorganize and extend the theory of metric spaces, suppressing the metric and emphasizing the collection of the open sets. There will be much talk of convergence of nets. In case you are still wary of nets, you can mitigate the culture shock by mentally substituting “sequence” for “net.” For the time being, the difference will be irrelevant. You will be fairly warned as soon as the nets become essential to the plot.

Curves in the Plane

So far, our arguments mainly dealt with analysis. In this chapter, we make a bigger step in the direction of topology. In fact, we will prove a famous topological theorem (Brouwer’s Fixed Point Theorem) and formulate a second one (Jordan’s Closed Curve Theorem). A complete proof of the latter will be given in Chapter 16.

Axioms for ℝ

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.