ADDED: Response to Minhyong's comment: I don't mean that any metric should actually be used. The whole point is that you want to study the properties of a space (here, a smooth manifold) without imposing a specific metric on the space. To me, the idea and definitions of topology arise naturally when you want to identify the properties of the space that remain valid, no matter which metric you use (changing the metric corresponds to stretching or warping the space).
ADDED:4) I see now that Minhyong's question is really about teaching a point-set topology course. I am ill-equipped to answer properly, because I have never taught such a course. I did take one at Penn, while I was an undergraduate, and I have to say that, although the course was taught very well, I found it even back then to be rather pointless. Nothing has changed my mind about this since then. For me, I find topology to be a compelling subject only when it is modified by adjectives such as "differential", "algebraic", "symplectic", "arithmetic" (?), but not when it is unmodified or modified by "point-set".