bigideas

Time for another mini-series on a mathematical topic, and this time we’re going to tackle the basics of topology. We’ll take a geometric approach in this series; I’d like to get up to the definition of the fundamental group but let’s see how we get on.

A topology, or topological space, is an idea with extremely widespread applications in mathematics; topologies arise in unexpected places, even in logic these days (where topos theory is helping physicists get to grips with the aparrent paradoxes of quantum mechanics). There are hard open problems in the heart of pure topology, too; recently Grisha Perelman claims to have proved the Poincare Conjecture; if he’s right then he’ll be eligible for one of the Clay Institute‘s million-dollar prizes, although he’s indicated he would turn it down.

It has wider applications too, though; simple topological ideas have been used by Achille Varzi, Mario Bunge, Barry Smith and others in attempts at modernising philosophical metaphysics. Without a grasp of the basic definitions, some important and rewarding parts of the philosophical literature are inaccessible.

In fact, the term “topology” covers a number of things. In the next post we’ll give the definition of a topological space, which is a very general and abstract definition. This opens the door to the topic of general topology, which has much of the same flavour as set theory. General topology mainly studies spaces that are very unlike our own; the line with two origins is a classic example. General topology also applies to familiar spaces, it just isn’t so interested in them.

We’ll have a quick look at some general topology, and exhibit a few curiosities, but we’ll move swiftly on to some geometric topology, in which the spaces are usually more like our own, only often with more dimensions (here are some nice examples). Geometric topology, beyond its most elementary form, tends to shade into algebraic topology; we’ll try, in this series, to get onto the first rung of that ladder by describing the “fundamental group”. It should be fun, if this sort of thing is your idea of fun.

This post is a part of a Topology Mini-Series