by Carl Pierer
Of the many properties a space can have, one of the most intuitively clear seems to be connectedness. Connectedness appears to be the simple property of hanging together in one piece. But how do we make this notion precise? On the one hand, we could think of it as the property that we can reach any point in the space without stepping outside. That is, in other words, that there is a path from any given point to any other point in the space. So, this would make the letter "o" connected, while the letter "i" wouldn't be. This notion corresponds to the topological notion of path connectedness. We say that a space is path connected if for any two points in the space there exists a continuous path from one to the other (which lies entirely inside the space).
On the other hand, we can think of connectedness in slightly different terms. The notion is perhaps clarest if we think about a musical melody. Somehow a succession of sounds turns into a melodic whole, which hangs together in a meaningful way. Breaking up the melody, interrupting it at any one point, can – provided the melody is complex enough – create two full, individual melodies, or one full melody and a somewhat incomplete, unresolved one, or two incomplete succession of notes. If the melody doesn't break into two simpler, shorter melodies, then at least one of the pieces will carry a certain tension that points beyond itself to a resolution of the harmonic build-up. If this is the case, we can think of the melody as in some way the simplest meaningful whole – there is no natural way to separate it into meaningful simpler pieces. This is the second notion of connectedness.
With some poetic licence, we can link the idea of an unresolved melody to the notion of openness, whilst a resolved melody would correspond to closedness, in topological terms. This identification has to be taken with a grain of salt, however, for the two notions – to a topologist – are not opposites. A set can be both open and closed or neither. What does justify the idea, however, is that an unresolved melody, much like an open set, points to a resolution that lies beyond itself: the tone that would form a complete whole, or the limit point that would render the set not open. In any case, we get the second notion of connectedness as a space that does not break up into two, non-empty and non-intersecting, open sets. To translate this into the language of melodies would be to say that a melody is connected if the only way to break it up is that there is one full, simpler melody and one unresolved.
Let us first consider the real number line. This is connected. To see why, let us try to split it into two disjoint open sets. This means choosing a real number ‘a' at which we cut the line in half. We can put one part of the real numbers (everything that is smaller than ‘a') on one side (A), and the other part (everything that is bigger than ‘a') on the other (B). This gives us two open sets in the sense suggested above: each of them is somewhat incomplete and points beyond itself – on one end (A) points to negative infinity, whilst on the other we can get as close to ‘a' as we like without ever reaching it, and similarly with (B). What to do with ‘a'? As soon as we stick it into either (A) or (B), one of the sets loses the property of pointing beyond itself. Doing so creates a hard border. Since ‘a' was chosen arbitrarily, this shows that we cannot separate the real line into two open sets and so it is connected.
Now, in most cases that we are familiar with, the two notions of connectedness coincide. In fact, however, path-connectedness is a stronger condition, that is, if something is path-connected then that implies it is also connected. It is not easy to see examples of spaces that are connected but not path-connected. One of these examples is the so called "Topologist's Sine curve". As a set, it is defined as:
{(x,y) in R2 | y = sin(1/x), x in (0,1]} ∪ {(0,0)}
Or in other words, the graph of the function f(x)=sin(1/x) for x in the interval (0,1] with the origin thrown in. In a picture this looks something like (Fig. 1). This space has the feature of being connected, but not path-connected.
First, we will check that it does not break up into two disjoint open sets. Let us begin with the point (0,0). According to our definition, a point is not an open set. This means that any open set that contains the point (0,0) will have to contain at least part of the graph of f(x). This graph, however, is much like the example of the real line discussed above. If we were to cut it at any one point ‘b', we would face the same problem of where to stick the point ‘b'. Thus we see that the space does not break up into two disjoint open sets and so the space is connected.
However, there is no continuous path from the origin to the graph. The rigorous notion of continuity lies beyond the scope of this brief exposition, but here it does coincide with the idea that a continuous change means not changing too abruptly. For instance, something like (Fig. 2) would be discontinuous, whilst any graph in R2 that can be drawn without lifting the pen from the paper would be continuous. Now, imagine we follow the graph part of our space from right to left. As we approach x=0, the graph starts to oscillate faster and faster. This oscillation keeps accelerating the smaller x gets; to point where at one instant we have f(x) = 1 and in the next instant, by only a minimal change in x, we have f(x) = -1. Since sin(z) is a continuous function, the graph is always continuous. If however, we wish to move from (0,0) (which is not part of the graph) onto the graph by even the tiniest change of x we would not know where we find ourselves on the graph. The oscillation just gets too fast. There can, thus, be no continuous path from (0,0) to the graph. This shows that the space is not path-connected.
The distinction between path-connectedness and connectedness is a subtle one. It does take some rather construed example, like the topologist's sine curve, to show that the latter is weaker than the former. However, perhaps there is a more intuitive way of understanding connectedness by likening it to musical melodies and how they hang together.