I encounter the topological notion of connectedness quite often lately, but I don't really have a good intuitive picture of it. How does one think of connectedness intuitively?
To provide some context, I do, in contrast, have an intuitive picture of path-connectedness. It simply makes sense to me to say that a space is (path-) connected whenever every two points can be connected by a path. Why then is the notion of connectednes prefered over path-connectedness? Is this only because of technical advantages (in the sense that they make it possible to prove more theorems), or is there also an intuitive motivation to extend the notion of path-connectedness to connectedness? And what are typical situations where this difference plays an important role, given the fact that for subsets of $\mathbb R^n$ and $\mathbb C^n$, for instance, the two notions are (EDIT: often) equivalent?