\nWe’re pretty much done with category theory, so it’s topology time!
\nSo what’s topology about? In some sense, it’s about the fundamental abstraction of *continuity*: if I have a bunch of points that form a continuous line or surface, what does that really mean? In particular, what does it mean *from within* the continuous surface?
\nAnother way of looking at is as the study of what kinds of *structures* are formed from continuous sets of points. This viewpoint makes much of topology look a lot like category theory: a study of mathematical structures, what they mean, and how we can build them and create mappings between them.
\nLet’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.
\nIt’s not just a joke; there’s a real example hidden in there. 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, and *how* they are connected. In the following diagram, all three shapes are *topologically* identical:
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![\"toruses.jpg\"](\"https:\/\/i0.wp.com\/scientopia.org\/img-archive\/goodmath\/img_36.jpg?resize=487%2C172\")
\nIf you turn the coffee mug into clay, you can remold it from mug-shape to donut-shape *without tearing or breaking*. 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 tearing a whole in it. If you’ve got a sphere and you want to turn it into a torus, you can either flatten it and punch a hole in the middle; or you can roll it into a cylinder, punch holes in the ends to create a tube, and then close the tube into a circle. And you can’t turn a torus into a sphere without tearing it: you need to break the circle of the torus and then close the ends to create a sphere. In either case, you’re tearing at least one whole in what was formerly a continuous surface.
\nTopology 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:
\n1. **Metric topology**: the study of *distance* in different spaces. The measure of distance and related concepts like angles in different topologies.
\n2. **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.
\n3. **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.)
\n4, **Network topology**: topology in the realm of discrete math. Network topologies are graphs (in the graph theory sense) consisting of nodes and edges.
\n5. **Differential Topology**: the study of differential equations in topological spaces that have the properties necessary to make calculus work.
\nPersonally, I find metric topology rather dull, and differential topology incomprehensible. Network topology more properly belongs in a discussion of graph theory, which is something I want to write about sometime. 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.
\nOne of the GM\/BM readers, Ofer Ron (aka ParanoidMarvin) is starting a new blog, called [Antopology][antopology] where he’ll be discussing topology, and we’re going to be tag-teaming our way through the introductions. Ofer specializes in geometric topology (knot theory in particular, if I’m not mistaken), so you can get your dose of geometric topology from him.
\n[antopology]: http:\/\/antopology.blogspot.com\/<\/p>\n","protected":false},"excerpt":{"rendered":"