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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.