When I took a poll of topics that people wanted my to write about, an awful lot of you asked me to write about topology. I did that before - right after I moved my blog to ScienceBlogs. But it's been a while. So I'm going to go back to those old posts, do some editing and polishing, correct some errors, and repost them. Along the way, I'll add a few new posts.
We'll start with the fundamental question: just what is topology?
I've said before that the way that I view math is that it's fundamentally about abstraction. Math is taking complex ideas, breaking down to simple concepts, and then exploring what those concepts really mean, and exactly what you can build using them.
I argue that topology, at its deepest level, is about continuity and nearness. In a continuous surface, what does in mean for things to be close to one another? What kind of structures can you build using nothing but nearness relationships? What is a structure defined solely by that notion of nearness?
When I say that continuity is fundamental to the basic abstraction of topoligy, some people - particularly topologists! - get upset, because continuity has a very specific meaning within topology, and that is not the meaning that I'm using when I make that statement.
In topology, you take a set of points, and you define their structure by which points are close to each other. Most of the basic topological structures are built using infinite sets of points, where there is no real notion of two points that are immediate neighbors; there's a continuum of ever smaller regions that define ever closer and closer neighborhood relationships. The kind of continuity that I'm talking about is more like the continuity of a surface - it's that fundamental notion of things that are close to each other, with that infinite ability to get take narrower and narrower subsets around a point. The kinds of structures that you get from doing that are interesting. In some sense, you're defining shapes - but they're malleable, squishy, twisty shapes, because all that matters is what points are close to what other points - not what direction you need to go to get from one to another.
Let's take a quick look at an example. There's a famous joke about topologists; you can always recognize a topology at breakfast, because they're the people who can't tell the difference between their coffee mug and their donut.
Like most good jokes, there's a kernel of truth hidden inside of it. From the viewpoint of topology, the coffee mug and the donut are the same shape. They're both toruses. In topology, the exact shape doesn't matter: what matters is the basic continuities of the surface: what is connected to what, what points are close to what other points. In the following diagram, all three shapes are topologically identical:
If you turn the coffee mug into clay, you can remold it from mug-shape to donut-shape without tearing it, or breaking it, or gluing any edges together. One can become the other by just squishing and stretching: so in topology, they are the same shape.
On the other hand, a sphere is different: you can't turn a donut into a sphere without punching a hole in it; and you can't turn a sphere into a torus without either punching a hole in it, or stretching it into a tube and gluing the ends together. You can't turn one into the other without changing the basic continuity of the shape.
To look at it slightly more formally: take a sphere. Now, punch a hole through it, to turn in into a torus. If you think about the points that surround the donut-hole, they used to be close - that is, neighbors of - the points on the other side of the hole. But after the hole is punched through, they're really far away - you need to go all the way around the hole to get to them, when they used to be right next door. So you've changed the nearness relations by making that hole.
Topology was one of the hottest mathematical topics of the 20th century, and as a result, it naturally has a lot of subfields. A few examples include:
- Metric topology
- the study of distance in different spaces. The measure of distance and related concepts like angles in different topologies.
- Algebraic topology
- the study of topologies using the tools of abstract algebra. In particular, studies of things like how to construct a complex space from simpler ones. Category theory is largely based on concepts that originated in algebraic topology.
- Geometric topology
- the study of manifolds and their embeddings. In general, geometric topology looks at lower-dimensional structures, most either two or three dimensional. (A manifold is an abstract space where every point is in a region that appears to be euclidean if you only look at the local neighborhood. But on a larger scale, the euclidean properties may disappear.)
- Network topology
- topology in the realm of discrete math. Network topologies are graphs (in the graph theory sense) consisting of nodes and edges.
- Differential topology
- the study of differential equations in topological spaces that have the properties necessary to make calculus work.
Personally, I find metric topology to be remarkably dull, so I'm not going to write much about it. Differential topology is completely beyond the capabilities of my puny brain, so there's no way that I can write anything intelligent about it. Network topology more properly belongs in a discussion of graph theory, which is something I've written about before. So I'll give you a passing glance at metric topology to see what it's all about, and algebraic topology is where I'll spend most of my time.