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Math Help - covering spaces

  1. #1
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    covering spaces

    Let X be a locally simply connected space. (X1, p1) is a cover for X. (X2,p2) is a covering space for X1. Then prove that (X2,p1*p2) is a covering space for X.

    I thought of this. But I am quite sure that I am wrong.

    Let x1 be an element X1 such that p(x1)=x. Then there is an admissible open set U1 is X1, such that p2^(-1) (U1) is a disjoint union of open sets each of which is homeomorphically mapped onto U1 by the map p2. I thought p1(U1) will be the required admissible open set about x. But I am not sure.
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  2. #2
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    Re: covering spaces

    What definition are you using for a covering?
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  3. #3
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    Re: covering spaces

    (X1,p) is a covering space for X if for each x in X, there exists an admissible open set U in X containing x, such that p^(-1) (U) is a disjoint union of open sets, each of which is homeomorphically mapped onto U by p. p should be surjective too.
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  4. #4
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    Re: covering spaces

    If X is locally simply connected then for each x\in X there exists a neighborhood U such that the map \pi_1(U,x)\rightarrow \pi_1(X,x) induced by the inclusion is trivial. Use this fact to first show that p_1^{-1}(U) is a disjoint union of open sets C_\alpha such that p_1 |_{C_\alpha} is a homeomorphism for all \alpha.

    By the way, semi-locally simply connected is sufficient for this result.
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