Hi, All: I am trying to understand why covering maps have the local triviality condition, i.e., given a cover C:X-->Y, every point y in Y has a neighborhood Oy of y with p^-1(Oy)~ Oy x F, where F is the fiber. This seems confusing, in that fibers of covering maps are a (discrete) collection of points in X (since local diffeomorphisms are local bijections, the preimages are isolated/discrete) . Does this statement just mean that the fiber is just a disjoint/discrete collection of open sets, indexed by the cardinality of the fiber, or is there more than that to it? I am thinking of standard examples like the covering map f:R-->$S^1$, given by f(t)=(cost,sint), an infinite-to-one cover; would the Oy here be any 'hood (neighborhood) of y that evenly-covers y? Also: if this local triviality holds: is every covering map a bundle of C over X with singletons as fibers? Sorry, I know this is simple, but I haven't seen it in a while, and hopefully someone's comments will jog my memory.