Because the display was getting quite cluttered, I thought I'd post a second part to this question separately. I hope the Gods of Math Overflow don't take too much offense. I'll go now into some details with which I wished intially to avoid prejudicing the replies.
Here are three natural areas of progression in the study of topology:
(1) point-set topology;
(3) sheaves and their cohomology;
and a very important
(2) middle ground
that I won't give a name to, involving matters like the classification of two-manifolds. I don't quite agree that the metric space intuition is sufficient even for a first course, simply because we always look ahead to the way the material will help or hinder students' future understanding.
Now, it is in (3) that a topology in the abstract sense plays a very active role, as the relation to global invariants emerges prominently. At this point, the definition needs no other motivation. However, in an introductory course on (1), I have yet to incorporate successfully interesting material from (3). When dealing with (2), curiously enough, the definition of a topology plays a very passive role. Many arguments are strongly intuitive. However, it is clear that one needs to have already absorbed (1) for the material in (2) to feel really comfortable. Of course there are exceptionallly intuitive people who can work fluently on (2) without a firm foundation in point-set topology. But for most ordinary folks (like me), the rigor of (1) is needed if only as a psychological crutch. Note here that most of the natural spaces that come up in (2) are in fact metrizable. However, getting bogged down with worrying about the metric would be a definite hindrance in working through the operations that come up constantly: stretching, bending, and perhaps most conspicuously, gluing. A quotient metric is a rather tricky notion, while a quotient topology is obvious.
With this future work in mind, how best then should one get this background in (1)? When I was an undergraduate, in fact, there was plenty of motivation in course (1). We used the first part of Munkres' book, and had exercises dug out of 'Counterexamples in Topology,' involving many strange spaces that have one property but not another. It was great fun. You may wonder then, what exactly I'm worried about. It's that I felt later that this kind of motivation had not been quite right. It wasn't entirely wrong either. Certainly we became very confident in working with the definitions, and that was good. However, and this is a big 'however,' when I moved on to (2), I had a distinct sense that the motivational material and exercises used in (1) were actually preventing me from learning the new notions efficiently. It took me quite a long time to make the transition, bugged by a persistent longing for the axioms and the exotic examples. A number of conversations I've had over the years indicate that my experience was far from unique. At any rate, this experience makes the issue for me quite different from a course on, say, linear algebra. No doubt the motivational material for diagonalization used in most courses is hardly convincing. Substantive examples come later, sometimes much later. But most of the operators and eigenvectors in standard textbooks are good toy models for a wide range of serious objects.
One obvious remedy would be to incorporate material from (2) into (1). I found this very hard, mostly because of the point already made above, that the role of (1) in (2) is quite implicit. When I first posed the question, I was hoping someone had figured out a good way of doing something like this. Incidentally, sheaves came around much later than the definition of a topology, so the historical question remains as well. How were the standard properties decided upon?
Let me make clear that I am not arguing that everyone has to go through the progression just outlined. Obviously, some people will take (1) elsewhere, in a way that the issues become quite different. Perhaps many people will never need more than metric spaces (or normed spaces, for that matter). But (1)-->(2)-->(3) is certainly common enough (perhaps especially for arithmetic geometers) to call for some reasonable methodology.
Meanwhile, I also appreciate Andrew Stacey's point (possibly even more than he does!). The long paragraphs above notwithstanding, the pedagogical question isn't something I lose sleep over. But it would be nice to have a few concrete and systematic ideas to use. They would certainly help me to understand the subject better!
Perhaps I should rephrase the question: How should we teach point set topology so as to facilitate the transition to the topology of natural spaces?
Somehow, this way of putting it seems much vaguer to me. 'How to teach X?' is such a broad question I would never be able to answer it myself in a finite amount of time. It seems to invite a diffuse discussion of everythng under the sun. That's why I had preferred to focus on one rather precise mathematical facet of that question. But I don't feel too strongly about it either way.