Previous: [[F_2 and the Banach-Tarski paradox]] Now that we're done with the group theory and far too many diversions into category theory, we can finally talk about the topology! One dumbed-down but not-entirely-incorrect way of looking at topology is that it "counts holes" of spaces to figure out when two spaces are the same or different. Intuitively, two homeomorphic spaces cannot have the same number of "holes." But what even is a hole? The most "obvious" kind of hole is the one you get by just removing a point. Consider the punctured plane, $\mathbb R^2 \setminus (0,0)$. Then the removed origin is one type of "hole." One way of detecting this hole is to have a loop around it, like the unit circle. Think of the unit circle as the curve defined parametrically by $(\cos 2\pi t, \sin 2\pi t)$ that starts at $(1, 0)$ at time $t=0$ and ends back at that point at $t=1$. Now, if we fix $(1, 0)$ as a basepoint and try to slightly deform this loop into any other loop, that loop *must* go around the punctured origin at least once. In fact, if you count right, it will go around the origin a sum total of exactly once counterclockwise. This is known as the "winding number" of this curve, and we will show over time that it is an invariant of the curve. The important thing here is that you cannot pull the loop tight without passing through the origin, which has been deleted. This line of reasoning is very important, and leads quite naturally to the idea of the fundamental group, which we will explore in detail. However, there are many other types of holes, which are detected by higher homotopy groups or higher homology groups. At some point it becomes unintuitive to think of them as "holes," but it is a great starting point. Another really interesting idea of algebraic topology, which ends up being linked to this idea, is that of a "covering space." Think of a 2D top-down shooter video game, where if you go off one side of the screen, you wrap back around and come back from the other side. How would you implement this? Well, one straightforward way is to just keep the player coordinates on $\mathbb R^2$ and then project onto the screen by taking the remainder of their coordinate and the height/width of the screen. In this case we say that $\mathbb R^2$ "covers" the game configuration space (which we'll see is actually a torus). We'll build up to these ideas rigorously starting with the next few posts. Additionally, I will write some other posts about topological constructions that are nice to know in general that will (hopefully) run in parallel. Next: [[Homotopies]]