topology
goodmath | November 6, 2006After my [initial post about manifolds](http://scienceblogs.com/goodmath/2006/10/manifolds_and_glue.php), I wanted to say a bit more about gluing.
You can form manifolds by gluing manifolds with an arbitrarily small overlap - as little as a single point along the point of contact between the manifolds. The example that I showed, forming a spherical shell out of two circles, used a minimal overlap. If all you want to do is show that the topology you form is a manifold, that kind of trivial gluing is sufficient, and it's often the easiest way to splice things together.
But there are a lot of…
goodmath | October 31, 2006Back in the early days of Good Math/Bad Math, when it was still at blogger, one of the most widely linked posts was one about the idea of *dimension*. At the time, I said that the easiest way to describe a dimension was as *a direction*. If you've got a point in a plane, and you want to say where it is, you can do it with two numbers - one for each of the fundamental directions in the plane. If you've set an origin, "(5,-2)" is enough to uniquely identify exactly one point. You con reach any point on the plane by moving in two directions: up/down and left/right.
If you've got a cube, you…
goodmath | October 26, 2006Time to get back to some topology, with the new computer. Short post this morning, but at least it's something. (I had a few posts queued up, just needing diagrams, but they got burned with the old computer. I had my work stuff backed up, but I don't let my personal stuff get into the company backups; I like to keep them clearly separated. And I didn't run my backups the way I should have for a few weeks.)
Last time, I started to explain a bit of patchwork: building manifolds from other manifolds using *gluing*. I'll have more to say about patchwork on manifolds, but first, I want to look at…
goodmath | October 17, 2006So, after the last topology post, we know what a manifold is - it's a structure where the neighborhoods of points are *locally* homeomorphic to open spheres in some ℜn.
We also talked a bit about the idea of *gluing*, which I'll talk about
more today. Any manifold can be formed by *gluing together* subsets of ℜn. But what does *gluing together* mean?
Let's start with a very common example. The surface of a sphere is a simple manifold. We can build it by gluing together *two* circles from ℜ2 (a plane). We can think of that as taking each circle, and stretching it over a bowl until it's…
goodmath | October 4, 2006Manifolds
So far, we've been talking about topologies in the most general sense: point-set topology. As we've seen, there are a lot of really fascinating things that you can do using just the bare structure of topologies as families of open sets.
But most of the things that are commonly associated with topology aren't just abstract point-sets: they're *shapes* and *surfaces* - in topological terms, they're things called *manifolds*.
Informally, a manifold is a set of points forming a surface that *appears to be* euclidean if you look at small sections. Manifolds include euclidean surfaces…
goodmath | October 2, 2006If you've got a connected topology, there are some neat things you can show about it. One of the interesting ones involves *fixed points*. Today I'm going to show you a few of the relatively simple fixed point properties of basic connected topologies.
To give you a taste of what's coming: imagine that you have two sheets of graph paper, with the edges numbered with a coordinate system. So you can easily identify any point on the sheet of paper. Take one sheet, and lay it flat on the table. Take the *second* sheet, and crumple it up into a little ball. No matter how you crumple the paper into…
goodmath | September 27, 2006Next stop on our tour of topology is the idea of *connectedness*. It's an important concept that defines a lot of useful and interesting properties of topological spaces.
The basic idea of connectedness is very simple and intuitive. If you think of a topology on a metric space like ℜ3, what connectedness means is, quite literally, connectedness in the physical sense: a space is connected if doesn't consist of two or more pieces that never touch.
Being more formal, there are several equivalent definitions:
* The most common one is the definition in terms of open and closed sets. It's…
goodmath | September 25, 2006This is going to be a short but sweet post on topology. Remember way back when I started writing about category theory? I said that the reason for doing that was because it's such a useful tool for talking about other things. Well, today, I'm going to show you a great example of that.
Last friday, I went through a fairly traditional approach to describing the topological product. The traditional approach not *very* difficult, but it's not particularly easy to follow either. The construction isn't really that difficult, but it's not easy to work out just what it all really means.
There is…
goodmath | September 21, 2006
One of the really neat things you can do in topology is play games with dimensions. Topology can give you ways of measuring dimensions, and projecting structures with many dimensions into lower-dimensional spaces. One of the keys to doing this is understanding how to combine different topologies to create new structures. This is done using the *topological product*.
So what's a topological product? It's almost the same thing as a cartesian product of sets - except, of course, that it needs to preserve the topological structure of open sets (and therefore neighborhoods). One way of saying…
goodmath | September 18, 2006Just like you can define a sub-set of a set, or a sub-object of an object in a category, you can define a sub-*space* of a topological space. It's a pretty easy thing to understand; interestingly, a sub-space of a topological space works in pretty much exactly the same way as a sub-sets and sub-object. In fact, the topological definition of a sub-space is *identical* to the categorical definition of a sub-object when we're looking at the category of topologies, **Top**.
Today, I'm going to explain what a subspace is, and show you how the categorical sub-object corresponds to the topological…
goodmath | September 11, 2006When we talk about topology, in general, the way we talk about it is in terms of *shapes*: geometric objects and spaces, surfaces, bodies that enclose things, etc. We talk about the topology of a *torus*, or a *coffee mug*, or a *sphere*.
But the topology we've talked about so far doesn't talk about shapes or surfaces. It talks about open sets and closed sets, about neighborhoods, even about filters; but we haven't touched on how this relates to our *intuitive* notion of shape.
Today, we'll make a start on the idea of surface and shape by defining what *interior* and *boundary* mean in a…
goodmath | September 6, 2006The past couple of posts on continuity and homeomorphism actually glossed over one really important point. I'm actually surprised no one called me on it; either you guys have learned to trust me, or else no one is reading this.
What I skimmed past is what a *neighborhood* is. The intuition for a
neighborhood is based on metric spaces: in a metric space, the neighborhood of a
point p is the points that are *close to* p, where close to is defined in terms of the distance metric. But not all topological spaces are metric spaces. So what's a neighborhood in a non-metric topological space?
A…
goodmath | September 5, 2006With continuity under our belts (albeit with some bumps along the way), we can look at something that many people consider *the* central concept of topology: homeomorphisms.
A homeomorphism is what defines the topological concept of *equivalence*. Remember the clay mug/torus metaphor from from my introduction: in topology, two topological spaces are equivalent if they can be bent, stretched, smushed, twisted, or glued to form the same shape *without* tearing.
The rest is beneath the fold.
*(A very important thing about the intuition above: understanding homeomorphism in terms of deforming…
goodmath | September 4, 2006*(Note: in the original version of this, I made an absolutely **huge** error. One of my faults in discussing topology is scrambling when to use forward functions, and when to use inverse functions. Continuity is dependent on properties defined in terms of the *inverse* of the function; I originally wrote it in the other direction. Thanks to commenter Dave Glasser for pointing out my error. I'll try to be more careful in the future!)*
Since I'm back, it's time to get back to topology!
I'm going to spend a bit more time talking about what continuity means; it's a really important concept in…
goodmath | August 24, 2006Yesterday, I introduced the idea of a *metric space*, and then used it to define *open* and *closed* sets in the space. (And of course, being a bozo, I managed to include a typo that made the definition of open sets equivalent to the definition of closed sets. It's been corrected, but if you're not familiar with this stuff, you might want to go back and take a look at the corrected version. It's just replacing a ≤ with a <, but that makes a *big* difference in meaning!)
Today I'm going to explain what a *topological space* is, and what *continuity* means in topology. (For another take on…
goodmath | August 23, 2006Topology usually starts with the idea of a *metric space*. A metric space is a set of values with some concept of *distance*. We need to define that first, before we can get into anything really interesting.
Metric Spaces and Distance
------------------------------
What does *distance* mean?
Let's look at a set, S, consisting of elements s1, s2, s3,...,sn. What does it mean to measure a *distance* from si to sj?
We'll start by looking at a simple number-line, with the set of real numbers. What's the distance between two numbers x and y? It's a measure of *how far* over the number-line you…
goodmath | August 22, 2006Back when GM/BM first moved to ScienceBlogs, we were in the middle of a poll about the next goodmath topic for me to write about. At the time, the vote was narrowly in favor of topology, with graph theory as a very close second.
We're pretty much done with category theory, so it's topology time!
So 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?
Another way of looking at is as the…
goodmath | August 15, 2006The Poincarė conjecture has been in the news lately, with an article in the Science Times today. So I've been getting lots of mail from people asking me to explain what the Poincarė conjecture is, and why it's a big deal lately?
I'm definitely not the best person to ask; the reason for the recent attention to the Poincarė conjecture is deep topology, which is not one of my stronger fields. But I'll give it my best shot. (It's actually rather bad timing. I'm planning on starting to write about topology later this week; and since the Poincarė conjecture is specifically about topology, it…