4 - Topology and uniformization

Summary

In the construction of Euclidean distance in Section 1.8, we define an atlas for our domain in which every point is contained in a puzzle piece, or chart, that looks like a small disk. The overlaps between the pages of the atlas are given by maps that identify parts of the charts on each page.

We use this idea to define smooth surfaces and in particular Riemann surfaces. We work in some generality but in most of the book our surfaces will be domains that sit inside the standard Euclidean plane.

We will assume that we have a well defined topology on our domains and surfaces. This means that, about each point p in the domain, there is a disk {z|z-p|<∈} contained in the domain. These disks form a basis for the open sets of the domain and they, in turn, satisfy standard conditions for unions and intersections. (See for example.) Unless otherwise stated, our domains are connected. This means that they cannot be expressed as the union of disjoint open sets.

We will need to be able to separate points in our domains. That is, we will assume that, given any two distinct points, we can find disjoint disks, each containing one of the points. A space with this property is called a Hausdorff space.

Surfaces

We are now ready to give precise definitions for our notions.

Definition 4.1A topological space is a set X with a collection of subsets O called its open sets. Both X and the emptyset belong to O.