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*.

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*.

Continue reading →