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Math Help - Covering spaces

  1. #1
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    Covering spaces

    Hello,

    I want to show, that any covering of the rectangle [0,1]^n is a trivial covering \forall n \in \mathbb{N}.
    I shall use the Homotopy Lifting thm and induction on n.

    I tried to solve this problem but i couldn't proceed very much.

    My first idea was the following:

    n=1:

    if we have a covering map p:Y->[0,1] , and a path in [0,1] for instance the identity path Id: [0,1]->[0,1]. for a given point y \in p^{-1}(0)
    there is a lifting of our path Id*:[0,1]->Y ,s.t. p \circ Id*=Id.

    we know that W:=Id*([0,1]) \subset Y is a connected subset and p(W)=[0,1].

    But why is p a trivial covering, i.e. why p^{-1}([0,1]) is a disjoint union of open sets, each homeomorphic to [0,1] under p??? I don't see it.

    I hope you can help me.

    Regards
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  2. #2
    Senior Member Tinyboss's Avatar
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    Do you know yet that universal covers are unique up to homeomorphism?
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  3. #3
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    No, i don't. We don't discuss universal covers yet.
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  4. #4
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    I could solve the problem for n=1. Does someone know how i can generalize, i.e. make the "induction step" (n-1)->n?

    Thanks a lot.
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