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    Number of connected components and continuous maps, Topology

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    The following question is answered:

    Let f:X-->Y be a continuous onto map. Let D be a subset of Y such that YD has at least n connected components. Prove that Xf^(-1)(D) has at least n connected components.

    Are the line and the plane with their usual topology homeomorphic?

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    Solution Preview

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    Let A_1,...,A_n be disconnected components of YD. We are done if we show that f^{-1}(A_1),...,f^{-1}(A_n) are
    disconnected in X. Indeed, as you can verify easily

    f^{-1}(YD)=f^{-1}(Y) f^{-1}(D)=X f^{-1}(D),

    since f^{-1}(Y)=X. It follows that since A_i is a subset of YD, then f^{-1}(A_i) is a subset of Xf^{-1}(D), and then under our assumption, f^{-1}(A_1),...,f^{-1}(A_n) are disconnected in X and so disconnected ...

    Solution Summary

    It is proved that the number of connected components of the inverse image of a set by a continuous onto map can not decrease. We use the result to answer the question of whether a line and a plane with their usual topologies are homeomorphic. The usual topology homeomorphic is examined.