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Math Help - Open and onto functions and bases

  1. #1
    DMT
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    Open and onto functions and bases

    I need some help with the following problem:

    Let f: (X,T) \to (Y,T*) be an open and onto function from the topological space (X,T) onto (Y,T*) and let \beta be a base for T. Prove that { f[ B ]: B \in \beta} is a base for T*.

    I understand that the image if any open set is open and that the image of an open set is the union of the images of the base elements of T. But since there is no continuity requirement Y can have an open set that has as an inverse image a set that is neither open nor closed and so isn't the union of the images of the bases elements of T.

    I think in fact the following trivial example shows this can't be true (which means I must be not understanding something fundamental):

    X=Y={a,b,c}

    Let T be the indiscrete topology on X, so that the only open sets are X and \emptyset. Then the only base for X is X itself.

    Define f: f(x)=x, but let T* be a different topology: say the open sets are X, \emptyset and {a}.

    f is still open an onto, but the image of all the base elements of T does not form a base for T*, since it misses {a}, which has the set {a} in X as an inverse image, which is neither open nor closed in (X,T).

    Any help telling me why my example is wrong and how to prove this would be appreciated.
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  2. #2
    MHF Contributor

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    Anytime that I have seen the question in textbooks, the function f has been given as continuous. You can see how a proof would go in that case?
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  3. #3
    DMT
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    It's easy if f is continuous. I was trying to figure out if somehow being open and onto implied continuous, but couldn't come up with anything.
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  4. #4
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    Quote Originally Posted by DMT View Post
    It's easy if f is continuous. I was trying to figure out if somehow being open and onto implied continuous, but couldn't come up with anything.
    Sorry I should have said more.
    I think that yours is a counterexample to that.
    BTW: After I posted, I found the problem stated exactly in GENERAL TOPOLOGY by Lipschutz. But I think is is a mistake. Mime is an edition from the 1960's. I tried to find an errata list somewhere but could not.
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  5. #5
    DMT
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    That's where I got the problem from. I've been working through the book on my own trying to give myself an overview of topology.
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  6. #6
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    Quote Originally Posted by DMT View Post
    That's where I got the problem from. I've been working through the book on my own trying to give myself an overview of topology.
    Actually I commend you for using that book for self-study.
    Are you particularly interested in general topology?
    I preface all this with one comment. I have taught topology to both undergraduates and graduate students. For undergraduates, I gave up on general topology texts such as Mendelssohn. I went to a problem-based approach (R.L. Moore is in my mathematical-genealogy) found in Elementary Theory of Metric Spaces by Reisel. I found that this approach gave students not only any overview of topology but a confidence by using a familiar idea of distance.
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  7. #7
    DMT
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    I'm interested in math in general, and topology was next on my list. Thanks for the other book recommendation, I might pick that one up. I thought I'd work through the book I have just to get a general feel for the topic, and then somewhere down the line maybe try and find time to take a class on the side.
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