@TylerLawson I have a question about your example of the coproduct of intervals of length one starting at each rational: "what" is the problem which prevents this map from being covering?
The example of the line with a double origin is an instance of gluing spaces along an open subset, which results in discrete fibers whose elements need increasingly smaller neighborhoods to be separated as we tend towards the boundary. On the other hand, this same problem of "fibers collapsing in the limit" can happen when we tend towards infinity, which (I think) is the role of properness in related ques…