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*.
- Log in to post comments
I haven't really seen the term "separated" used as much as "non-connected" or "disconnected". Actually, calling a non-connected space "separated" runs the risk of confusion with "separable", which is a technical analytic notion having almost nothing to do with this topological notion.
Wouldn't any connected topology necessarily also be path-connected?
Exactly why is it useful to maintain different definitions for connectedness and path-connectedness?
Coin:
No, connected topologies are *not* necessarily path connected. The standard textbook example is called the topologists sine curve: y = sin(1/x). Using the topology imposed on this by the standard topology for ℜ2, the topologists sine curve *is* connected. But it is *not* path connected.
I'm having trouble with your first and third definitions of connectedness. The open unit ball at the origin in R^n is clearly a connected topological subspace, but it's boundary is decidedly non-empty, and the open ball is definitely not closed. How does one apply those three definitions to subspaces? What am I missing? (My favorite topology book is at work, or I might look this up myself.)
I can see that _two_ not-separated subspaces are connected, but not how to test if a single subspace is connected (using a generalization of 1 or 3 to subspaces). Perhaps the second definition comes to our rescue by saying that a subspace is connected iff it's not the union of two disjoint subspaces? But then your defs 1 and 3 can't be obviously generalized to subspaces.
Thoughts?
You actually want the closure of the topologist's sine curve. Just y=sin(1/x) is disconnected if you're allowing negative x, or it's path connected if you restrict to positive x.
billb:
Those definitions are equivalent for topological spaces, *not* topological *sub*spaces. If you have the unit ball around the origin as a *full* space - i.e., there are no points outside of it - then it's closed.
Billb, you want to forget that a subspace lives in some bigger space, and just treat it as all there is. So if we look at the open ball, it has no boundary, because it is everything. Similarly, because it is everything, it is both open and closed.
This isn't what tells us that the ball is connected, though. To know that it's connected, we need to know that there are no subsets of the ball that are both open and closed (in the subspace topology). Since this is a negative condition, it can be rather hard to check. Perhaps the easiest way to show the unit ball is connected would be to show that it is path connected. Given any two points x and y in the ball, the straight line between them is a path that connects them, and this path lives entirely within the ball.
What you are calling a "separated" space is (almost) always called a DISCONNECTED space. I've never heard the terminology "separated space". I'm sure it prob. shows up in some textbook somewhere, but it's highly non-standard terminology.
"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."
How can a pair of subspaces be connected?? Connectedness by definition is a property of a space intrinsically. What you mean to say, I think, is:
"A top. space X is connected iff the only separated pair of subsets of X whose union is X is {X, empty}."
(Sometimes, a "separation" is defined to be a pair of non-empty separated subsets that partition the space. Then connected just means that no separation exist.)
Darin:
The term "separated space" may not be the most common one, but it is a valid term; it's in the textbook I used as a reference; it also appears in articles on Wolfram's mathworld and wikipedia.
And do you really mean to claim that saying a pair of subspaces is connected is meaningless? Give me a break...
You can play obnoxious pedant all you want; you can make meaningless corrections of informal prose all you want; it's not going to change the fact that *Duesberg was wrong*.
I have just been informed that "separated space" is actually synonymous with "Hausdorff space", which is something completely different. Although almost everyone says "Hausdorff".
"And do you really mean to claim that saying a pair of subspaces is connected is meaningless? Give me a break..."
YES, I do. Connectedness is an intrinsic property of a single space. Unless you mean to say "These two subspaces are each individually connected..." but who cares about that???
"You can play obnoxious pedant all you want; you can make meaningless corrections of informal prose all you want; it's not going to change the fact that *Duesberg was wrong*."
You stepped into the HIV debate by impugning the intelligence and integrity of dissidents publicly on this blog. You are hardly in a position to get on my case for pointing out how sutpid YOU are in math.
Darin:
Thanks for proving my point. You can play the offended pedant all you want; it will *never* change the fact that Duesberg made a pathetic, incompetent error.
(I don't think that anyone who wasn't being deliberately obnoxiously pedantic would disagree with a statement that a pair of subspaces is connected if/f the union of the two subspaces forms a connected topological subspace.)
Almost... you need to add a point on the y-axis for that to work. The set A U B, where A = {(0, y): -1 <= y <= 1} and B = {(x, sin(1/x): x > 0}, is connected but not path-connected.
However, it's fairly simple to prove that a topological manifold (in particular, any open subset of R^n) is connected iff it's path-connected.