Math 117: Topology of the Real Numbers

This is a very abstract definition, so for the time being the reader may want to focus instead on the notion of open sets within R, which will be developed in the following pages. The family T of open sets in R is the simplest example of topology. Gradually, it will become apparent that this topology is useful for formulating the various notions of limit, and we will see how these limits are used to define continuous functions. Ultimately, they are also used to define derivatives of functions. In this installment of the lecture notes, we highlight the most important topics presented in Chapter 3 of the text [1], and provide a few additional topics on metric spaces, in the hopes of providing an easier transition to more advanced books on real analysis, such as [3]. The study of the various topologies that one can define on sets has developed into an important branch of mathematics in its own right, called topology [2].