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Math Help - Totally lost with Sorgenfry

  1. #1
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    Totally lost with Sorgenfry

    Are the Rational numbers an open subset of the Real numbers with respect to the Sorg. topology?

    No really sure how to begin. Can somebody start me off?
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  2. #2
    Senior Member Tinyboss's Avatar
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    An open set is a union of basis elements. Is there any basis element of that topology that doesn't contain any irrational numbers?
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  3. #3
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    Well, am I correct in saying that this is the basis:

    B=\{ [ a,b):a \in R, b \in Q, a < b\}
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  4. #4
    Senior Member Tinyboss's Avatar
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    Surely you have the definitions available in your textbook or lecture notes?
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  5. #5
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    No, lecturer is the worst. I'm having a hard enough time finding info off of google without it being way too complicated.
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  6. #6
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    okay here is a basis, correct?

    \{[a,b) \times [c,d) : a, b,c,d \in R, a<b,c<d\}

    So, in answer to your question, all of the intervals contain irrational numbers?
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  7. #7
    Senior Member Tinyboss's Avatar
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    Now you seem to be putting a topology on R^2 (Sorgenfrey plane). What is a rational number in R^2? Do you mean elements of QxQ? If so, then the original hint applies and yes, your answer to the question is correct. Can you finish from there?
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  8. #8
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    To be absolutely honest, I don't know what to do next.
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  9. #9
    Senior Member Tinyboss's Avatar
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    If every set in a collection contains non-rationals, then any union of those sets must contain non-rationals. Every open set is such a union, and so...?
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  10. #10
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    and so the set of rationals cannot be an open subset?
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