A long time ago – in 2006! – I wrote a ton of blog posts about topology. In the course of trying to fix up some of the import glitches from migrating this blog to its new home, I ended up looking at a bunch of them. And… well… those were written in my early days of blogging, and looking back at those posts now… well, let’s just say that my writing has come a long way with 8 years of practice! I was thinking, “I could do a much better job of writing about that now!”
So that’s what I’m going to do. This isn’t going to be reposts,
but rather completely rewrites.
Topology is typical of one of the methods of math that I love: abstraction. What mathematicians do is pick some fundamental concept, focus tightly on it, discarding everything else. In topology, you want to understand shapes. Starting with the basic idea of shape, topology lets us understand shapes, distortions, dimensions, continuity, and more.
The starting point in topology is closeness. You can define what a shape is by describing which points are close to which other points. Two shapes are equivalent if they can be built using the same closeness relationships. That means that if you can take one shape, and pull, squash, and twist it into another shape – as long as you don’t have to either break closeness relations (by tearing or punching holes in it), or add new closeness relations (by gluing edges together) – the two shapes are really the same thing.
This leads to a very, very bad math joke.
How do you recognize topologists at breakfast?
They’re the people who can’t tell their donut from their coffee.
The easiest way to ruin a joke is to over-explain it. Happily, when the joke is this bad, it’s already ruined, so this isn’t my fault. See, in topology, a coffee mug is the same shape as a donut. They’re each three dimensional shapes with one hole. If you had a donut made of an infinitely stretchable material, you could shape it into a coffee mug without every tearing it, ripping a hole, or gluing two edges together.
Anyway – starting tomorrow, I’ll be posting a new version of that old topology series.