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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".
    Post Made Community Wiki by Anton Geraschenko♦♦
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Here are my reactions to the latest:

1) I think for the vast majority of people who work in differential topology and geometry and global analysis, the idea that topology expresses the notion of "closeness" without using a specific choice of a distance function is enough. In fact, for most of us, everything is locally Euclidean. This is certainly good enough for classifying 2-dimensional surfaces, which leads to my second thought...

2) I don't know what you mean by the middle ground. Could you elaborate? My ignorance is almost certainly due to 1).

3) It seems to me that, Cech cohomology notwithstanding, the progression to sheaves is critical only if the topological space has nontrivial local structure to it. My impression is that sheaves play a central role in subjects such as complex analytic, algebraic, and arithmetic geometry, because singularities are unavoidable yet tractable. They are seen much less in the smooth real category, where singularities are usually avoided and, despite the efforts of Thom, very difficult to deal with.
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