Sorry about the horrible confusion in this post, I am befuddled and likely spouting gibberish. Consider the general case of a one parameter family of metrics, where the parameter in question need not be finite (it can limit to infinity for instance). I am looking for a topology that is big enough to accomodate the entire family. (im not even sure how this should work, since im used to thinking off one topology for one metric) It seems to me in general, whatever this huge topology is, it need not and probably cannot be metricizable. Is this true, and can you think of any other constraints?