# Local Trivialization in Covering Spaces

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.

## Answers and Replies

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quasar987
Homework Helper
Gold Member
If Oy is evenly covered, then p^{-1}(Oy) is a disjoint union of sets homeomorphic to Oy and there are N of them, where N is the number of sheets of the covering (not necessarily an integer). But it seems indeed that p^{-1}(Oy) can also be seen as a product Oy x S, where S is any set of cardinality N equipped with the discrete topology, thus making a covering space into a locally trivial fiber bundle with discrete fibers.

mathwonk
Homework Helper
they have it so you can lift maps of curves into the base.

WWGD
Gold Member
2019 Award
Hmm.. that is interesting; every covering space is (can be made into) a bundle with discrete fiber. Is the opposite also the case, that every bundle is a covering space? It would seem so, but then what is the difference between the two?

quasar987
Homework Helper
Gold Member
Hmm.. that is interesting; every covering space is (can be made into) a bundle with discrete fiber. Is the opposite also the case, that every bundle is a covering space? It would seem so, but then what is the difference between the two?
Do you mean that or "is every bundle with discrete fiber a covering space"?

lavinia