So, after a few brief diversions, let’s get back to talking about metric spaces. Now that we’ve got the concept of open sets out of the way, we’ll be able to talk about some more interesting properties.
Today, we’ll be discussing the notion of connectedness. Much as was the case with open and closed sets, this is not actually a property unique to metric spaces: indeed, so long as we have open sets we can reasonably discuss connectedness. As we’ll see down the track though, connectedness is required in a lot of rather nice theorems that arise from the study of metric spaces.
Connectedness is motivated in a rather obvious way by our desire to be able to put a finger on when two things are ‘joined together’ or separated from each other. We can easily demonstrate what we’d like using circles (as is frightfully often the case in maths). So, first, let’s meet our circle:
Yep. That sure is a circle. Notice that I’ve given it a dashed border — that’s a pretty standard way to signify that the circle (or any other shape) is open.
One circle is pretty lonely, however, so let’s give him a friend:
Things are heating up now — we’ve got two circles. Welcome to the world of high-stakes maths, where anything can happen!
The circles still seem kind of sad, however, separated by a tiny strip of white. When we define connectedness in a short spell, we’re going to want to say that those two circles are disconnected.
So then, how can we make our maudlin circles perk up a little?
Avert your eyes, you filthy perverts! That pornographic display of interrupted periodicity is what we’re going to want to label a connected set.
A quick technicality before I give the formal definition, just to avoid confusion later down the track: The property of being connected or disconnected is something that an individual set has. So, when I say that the two non-overlapping circles above are disconnected, what I really mean is that the set consisting of two non-overlapping circles is disconnected. Each circle, treated individually, is a connected set.
So, formally, what is a connected set? We’ll define it in opposition to being a disconnected set:
A set is disconnected if it can be written as the union of two sets, , where and are both open and (i.e. the sets have no common elements).
With that in mind, we can now easily define a connected set:
A set is connected if and only if it is not disconnected.
Further study of this idea leads to a whole mess of different (and stronger) ways in which sets can be connected. The notion of path connectedness, for example, requires that given any two points in my set there is a path that joins them together. If a set is path connected it is also connected; the converse statement is not true, however — the topologist’s sine curve being a standard example (that I might discuss at a later date).
Filed under: Maths