Before we can talk about topological invariants we need to know two things: 1) what is a topology, and 2) what is an invariant. Simply put, a topology is a collection of sets, each of which, vaguely speaking, has fuzzy edges. An invariant, on the other hand, is basically something which doesn’t change. In the remainder of this article I will slowly introduce the concept of a topological invariant and then having done so will illustrate what you can do with them.
To properly define a topology let’s start with a non-empty set. We can, of course, give it a name: and it will consist of one or more objects, or subsets. Suppose further that we collect up a bunch of these subsets in a very particular way (explained in a second) and call it the Greek letter tau, or .
If the way in which we pick our subsets adheres to the following rules (or axioms):
- The empty set and the entire set are in
- Any union (possibly infinite) of any open set is in .
- The intersection of any finite number of the open sets is, again, open.
then the collection of subsets gets a special name, of course, it is called a topology on the set . The objects within this special kind of collection get their own name too, we call them open sets. Open sets are nice sets; we can pick elements out of them without having to worry about accidentally going too far and picking an element that isn’t in it. Formally speaking, an open set does not contain any of its boundary points and this gives us freedom to draw little circles around each and every element inside an open set and know for a fact that the little circle (also called a neighbourhood) is entirely within the set. Taking the set that we began with and the collection of subsets we form the tuple , and this is called a topological space.
A topological space thus consists of some set along with a collection of subsets that are nice and fuzzy. All in all, this is a pretty big space, almost all of mathematics takes place in some topological space or another. Topological spaces allow you to, obviously define sets and open subsets, which in turn allow you to define neighbourhoods which carry a diameter of sorts, which in turn allows you to define a primitive notion of distance (think of everything being measured in terms of the diameter of the little circles, or as it’s denoted). A notion of distance gives rise to the 3 C’s: Continuity, Connectedness and Convergence. And this, my friends, is where calculus lives.
The second concept we need is that of an invariant. An invariant is a class of mathematical objects that don’t change when you move the object out of one environment and in to another (or back to the original one). Technically speaking, moving an object doesn’t really happen, what actually happens is that you move the object’s elements and the way in which that is done is by use of a mapping, usually denoted by . The mapping takes an element, say of some set , and maps it to another element in some other set . Once you do this to each and every element of , if the destination set shares a property with then we say that the original set is invariant under .
Let’s begin by talking about an invariant of a homeomorphism. In case you don’t recall, a homeomorphism is a mapping (a way of getting from point A to point B), call it , from a topological space to another topological space that preserves the topological properties of the first space.