Introduction to Mathematical Topology
In order to understand the complexity of Riemann Surfaces, it is important to first understand some aspects of mathematical topology as well as some of the vocabulary that will be often associated with Riemann Surfaces. Once these terms are better understood, explaining Riemann Surfaces with this terminology makes this topic much more comprehensive.
One of the important terms when studying topology is the term “set” or “sets”. More specifically “open” sets, “closed” sets, “subsets”. In an open set, one should be able to choose any point within this set, take a small step in any direction he choses, and then immediately find another point within the same set. Some of the properties of an open set are that the whole space that takes up the set (which we will refer to as X) is open, “...the union of any collection of open sets is open, [and] the intersection of any finite number of open sets is open,” (Babinec et al., 2007). Open sets differ from closed sets by the fact that open sets are not bound by what are called limit points, points that essentially put boundaries on a set. A limit point is a point at which its neighborhood contains at least one point that is not in the set. Closed sets are bound by these limit points.
Another important pair of terms to know, especially now that a set has been explained, are topological spaces and topology. A topological space is a space without any notions of distance between two points, meaning the two points exist in space, but we do not know exactly where they are located in relation to each other in this arbitrary space. A topological space is defined by two values: X, which is a set, and τ, which we call a topology and is a set of subsets within the set X. However, a topology must satisfy the following requirements: the topological set must contain the empty set ∅ and the set X; it contains the union of any subsets in the topology; any intersections in the topology must be a part of the topology. For example, if a Є X and X is a set, then a basic topology for this would be: τ = {∅, X, {a}} If there were any additional subsets, lets say b, then the set would contain a, b,and the union of this set which would be {a,b}. Also an important topological idea is that of the Hausdorff Space. A Hausdorff Space is a space where there are two points that have neighborhoods that do not overlap. The image below gives a visual example of what a Hausdorff Space is. X and Y are the two separate neighborhoods around points a and b respectively. The neighborhoods do not intersect, and therefore they form a Hausdorff Space.
The last important term for understanding Riemann Surfaces is knowing what a homeomorphism is. Breaking this word into its Latin roots, “homeo-” means “same” or “alike” and “morph-” means “shape”. So, roughly, a homeomorphism means “same shape”, which makes sense because a homeomorphism is a way to describe turning one topological space into another to analyze better, but the space does not lose its original integrity and the spaces are essentially the same. Another way to think about this idea is when you transform a 2-D space that is misshapen into a different 2-D space that is more comprehensive (e.g. turning a rhombus-looking space into a square. They are essentially the same space, but the original has been transformed into something to analyze more easily). Figure 2 (“Homomorphism…”, 2014) below gives a common homomorphism example where a “doughnut”, formally known as a torus, can be bent, twisted, and shaped into a coffee cup. The spaces are the same, but they have just been shaped differently.
All of these terms and definitions in topology are crucial to understand the complexity of Riemann surfaces, specifically because these terms are all used in the definition of a Riemann Surface. Another idea that is important to understand, but so complex and important that it deserves its own section, is that of manifolds, which is also part of the Riemann Surface definition.
