Time 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 another way of building interesting manifolds.
At heart, I’m really an algebraist, and some of the really interesting manifolds can be defined algebraically in terms of topological product. You see, if you’ve got two manifolds **S** and **T**, then their product topology **S×T** is also a manifold. Since we already talked about topological product – both in classic topological terms, and in categorical terms, I’m not going to go back and repeat the definition. But I will just walk through a couple of examples of interesting manifolds that you can build using the product.
The easiest example is to just take some lines. Just a simple, basic line. That’s a 1 dimensional manifold. What’s the product of two lines? Hopefully, you can easily guess that: it’s a plane. The standard cartesian metric spaces are all topological products of sets of lines: ℜⁿ is the product of *n* lines.
To be a bit more interesting, take a circle – the basic, simple circle on a cartesian plane. Not the *contents* of the circle, but the closed line of the circle itself. In topological terms, that’s a 1-sphere, and it’s also a very simple manifold with no boundary. Now take a line, which is also a simple manifold.
What happens when you take the product of the line and the circle? You get a hollow cylinder.

What about if you take the product of the circle with *itself*? Thing about the definition of product: from any point *p* in the product **S×T**, you should be able to *project* an image of
**S** and an image of **T**. What’s the shape where you can make that work right? The torus.

In fact the torus is a member of a family of topological spaces called the toroids. For any dimensionality *n*, there is an *n*-toroid which the the product of *n* circles. The 1-toroid is a circle; the 2-toroid is our familiar torus; the 3-toroid is a mess. (Beyond the 2-toroid, our ability to visualize them falls apart; what kind of figure can be *sliced* to produce a torus and a circle? The *concept* isn’t too difficult, but the *image* is almost impossible.)

So, 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 shaped like a hemisphere. Then we glue the two hemispheres together so that the *boundary* points of the hemispheres overlap.
Now, how can we say that formally?

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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 – like the standard topology on a plane; but they also include many non-euclidean surfaces, like the surface of a sphere or a torus.

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If 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 a ball, no matter where you put it down on the uncrumpled sheet, there will be at least one point on the crumpled ball of paper which is directly above the point with the same coordinate on the flat sheet.
As a quick aside: today is Yom Kippur, which means that this post is scheduled, and I’m not anywhere near my computer. So I won’t be able to make any corrections or answer any comments until late this evening.

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Next 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 precise, concise, and formal; but it doesn’t have a huge amount of intuitive value. A topological space **T** is connected if/f the only two sets in **T** that are both open *and* closed are **T** and ∅.
* The most intuitive one is the simplest set based definition: a topological space **T** connected if/f **T** is *not* the union of two disjoint non-empty closed sets.
* One that’s clever, in that it’s got both formality and intuition: **T** is connected if the only sets in **T** with empty boundaries are **T** and ∅.
Closely related to the ida of connectedness is separation. A topological space is *separated* if/f it’s not connected. (Profound, huh?)
Separateness becomes important when we talk about *subspaces*, because it’s much easier to define when subspaces are *separated*; and they’re connected if they’re not separated.
If A and B are subspaces of a topological space **T**, then they’re *separated in **T*** if and only if they are disjoint from each others closure. An important thing to understand here is that we are *not* saying that their *closures* are disjoint. We’re saying that A and B^* are disjoint, and B and A^* are disjoint, not that A^* and B^* are disjoint.
The distinction is much clearer with an example. Let’s look at the topological space ℜ². We can have *A* and *B* be *open* circles. Let’s say that *A* is the open circle centered on (-1,0) with radius one; so it’s every point whose distance from (-1,0) is *less than* 1. And let’s say that *B* is the open circle centered on (1,0), with radius 1. So the two sets are what you see in the image below. *A* is the shaded part of the green circle. The outline is the *boundary* of *A*, which is part of *A^**, but not part of *A* itself. *B* is the shaded part of the red circle; the outline is the boundary. Neither *A* nor *B* include the point (0,0). But both the *closure* of *A* and the *closure* of *B* contain (0,0). So *A* is disjoint from *B^**; and *B* is disjoint from *A^**. But *A^** is *not* disjoint from *B^**: they overlap at (0,0). *A* and *B* are separated, even though their closures overlap.

There’s also an even stronger notion of connectness for a topological space (or for subspaces): *path*-connectedness. A space **T** is *path connected* if/f for any two points x,y ∈ **T**, there is a *continuous path* between x and y. Of course, we need to be a bit more formal than that; what’s a continuous path?
There is a continuous path from x to y in **T** if there is a continuous function *f* from the closed interval [0,1] to **T**, where *f(0)=x*, and *f(1)=y*.