Why do introductory real analysis courses teach bottom up? A big part of introductory real analysis courses is getting intuition for the $\epsilon-\delta\,$ proofs. For example, these types of proofs come up a lot when studying differentiation, continuity, and integration. Only later is the notion of open and closed sets introduced. Why not just introduce continuity in terms of open sets first (E.g. it would be a better visual representation)? It seems that the $\epsilon-\delta$ definition would be more understandable if a student is first exposed to the open set characterization.  

I'm with Alex Becker, I first learned convergence of sequences, using epsilon and deltas, and only later moved on to continuity of functions. It worked out great for me. I don't believe that the abstraction from topology would be useful at this point. The ideas of "$x$ is near $y$", "choosing $\epsilon$ as small as you want", etc, are better expressed by epsilon-delta arguments, because they quantify/translate the words "near" and "small".
Maybe one could talk about "size of intervals", grasping the idea of "open neighborhood" and retaining the epsilons. 
