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Math Help - [SOLVED] Continuous Functions/Maps

  1. #1
    is up to his old tricks again! Jhevon's Avatar
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    [SOLVED] Continuous Functions/Maps

    I need help with what I am sure just a simple tweek in concept can fix.

    In topology, we define a continuous map as follows:


    Definition: Let X and Y be topological spaces. A map f: X \to Y is called continuous if the inverse image of open sets is always open. (says the book, "Topology" by Klaus Janich)

    That is, (and this is not in the book) f: (X_1, Y_1) \to (X_2,Y_2) is a continuous map if for every set U \in Y_2 we have f^{-1}(U) \in Y_1




    My question is, how can we prove that this definition of continuity is equivalent to the calculus \epsilon - \delta definition of continuity?

    I'm pretty sure I have to say something about "metric spaces" and "neighborhoods" but there is a small conceptual leap i can't make to connect it all.

    If necessary, I can supply any definition I am working with upon request, but I don't want someone to write the proof for me, but in layman's terms, tell me how I should think about it, what conceptual links should I be making between calculus and topology in this respect?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Jhevon View Post
    I need help with what I am sure just a simple tweek in concept can fix.

    In topology, we define a continuous map as follows:


    Definition: Let X and Y be topological spaces. A map f: X \to Y is called continuous if the inverse image of open sets is always open. (says the book, "Topology" by Klaus Janich)

    That is, (and this is not in the book) f: (X_1, Y_1) \to (X_2,Y_2) is a continuous map if for every set U \in Y_2 we have f^{-1}(U) \in Y_1




    My question is, how can we prove that this definition of continuity is equivalent to the calculus \epsilon - \delta definition of continuity?

    I'm pretty sure I have to say something about "metric spaces" and "neighborhoods" but there is a small conceptual leap i can't make to connect it all.

    If necessary, I can supply any definition I am working with upon request, but I don't want someone to write the proof for me, but in layman's terms, tell me how I should think about it, what conceptual links should I be making between calculus and topology in this respect?
    In a metric space a set S is open if for every point in the set there is an open ball centred on the point which is a subset of S.

    The \epsilon - \delta definition of continuity is a definition in terms of open balls (which are themselves open sets). But an arbitary union of open sets is open, which should be sufficient to complete a proof that the \epsilon - \delta definition of continuity is equivalent to the topological defintion for the usual topology on \mathbb{R}.

    RonL

    Note: the \epsilon - \delta definition of continuity is something like: The inverse image of an open ball contains an open ball.
    Last edited by CaptainBlack; July 22nd 2007 at 11:31 PM.
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    In a metric space a set S is open if for every point in the set there is an open ball centred on the point which is a subset of S.

    The \epsilon - \delta definition of continuity is a definition in terms of open balls (which are themselves open sets). But an arbitary union of open sets is open, which should be sufficient to complete a proof that the \epsilon - \delta definition of continuity is equivalent to the topological defintion for the usual topology on \mathbb{R}.

    RonL
    i believe i see what you are saying, and i kind of had similar thoughts, but somehow i don't see what inverse mappings have to do with this. that's the link i was looking for. i want to explicitly tie the relationship between a continuous function and it's inverse into all this, and make similar ties with topological spaces
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by Jhevon View Post
    i believe i see what you are saying, and i kind of had similar thoughts, but somehow i don't see what inverse mappings have to do with this. that's the link i was looking for. i want to explicitly tie the relationship between a continuous function and it's inverse into all this, and make similar ties with topological spaces
    The \epsilon - \delta definition may be rephrased as the inverse image of an open ball centred on f(x_0) contains an open ball centred on x_0. Then the arbitary union of open sets property gives the result that an \epsilon - \delta continuous function is a continuous function wrt the usual topology on the reals.

    RonL
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