A first tour of topology: metric spaces
The most natural way one can introduce topology, and what topology is about, is to explain what a metric space is. A metric space is a set, X, equipped with a metric d, which serves to measure distances between points.
The purpose of this chapter is to introduce the basic notions that we shall explore throughout this book, in this more familiar setting.
However, we must face a conundrum. Imagine we explained what an opensubset is in a metric space. In the more general settings we shall explore in later chapters, we would have to redefine opens. Redefinitions are bad mathematical practice, and for a good reason: we would never know which definition we would mean later on; i.e., are the opens we shall ...