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Math Help - Continuous Functions and Topologies

  1. #1
    Super Member Aryth's Avatar
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    Continuous Functions and Topologies

    We have a definition for continuity in terms of open sets of topologies, but I had a question about it. Here's the definition:

    Let (X,\Omega) and (Y,\Theta) be topological spaces. A function f: X \to Y is continuous relative to \Omega and \Theta provided f^{-1}(U) \in \Omega for every U \in \Theta.

    My question is this: Is f^{-1} : Y \to X continuous relative to \Theta and \Omega provided U \in \Theta for every f^{-1}(U) \in \Omega? Or is there something else you have to do to that sentence to make it true?
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    Re: Continuous Functions and Topologies

    Quote Originally Posted by Aryth View Post
    We have a definition for continuity in terms of open sets of topologies, but I had a question about it. Here's the definition:
    Let (X,\Omega) and (Y,\Theta) be topological spaces. A function f: X \to Y is continuous relative to \Omega and \Theta provided f^{-1}(U) \in \Omega for every U \in \Theta.

    My question is this: Is f^{-1} : Y \to X continuous relative to \Theta and \Omega provided U \in \Theta for every f^{-1}(U) \in \Omega? Or is there something else you have to do to that sentence to make it true?
    This is a case in which the function notation is getting into the way of understanding.

    You have a function f:X\to Y.
    But then f^{-1} maps {P}(Y)\to {P}(X), i.e. between power sets.

    Continuity is defined on topological spaces. What is the topology on {P}(Y)~?
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    Re: Continuous Functions and Topologies

    The very first time I was asked to present a proof in a Graduate school class, it was precisely a problem involving f^{-1}(A) where A was in the image of f. I did the whole problem assuming that f was invertible (they said " f^{-1}" didn't they?)!
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