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Math Help - limit points of Aintersect B, closure of (AintersectB)

  1. #1
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    limit points of Aintersect B, closure of (AintersectB)

    I was given a problem from my book.
    a) If y is a limit point of AUB, show that y is either a limit point of A or a limit point of B.
    b) Prove that (AUB) closure=A closureUBclosure
    c) Does the result about closures in b)extend to infinite unions in sets


    I was basically asked to consider this problems with intersections instead of unions.



    a)Let y be a limit pt of AintersectB
    Then y=limy_n
    y_n is an element of A intersect B
    So some subsequence (y_n) must lie in A and B
    So y is a limit pt of A and B
    b)I will consider (AintersectB)closure=Aclosure intersect Bclosure and want to see if this is true.
    (AintersectB)closure=AintersectBUL(AintersectB)
    Part a says L(AintersectB)=L(A)intersectL(B)
    So (AintersectB)closure=A intersect BU L(A) intersect L(B)
    =(AUL(A)) intersect (BintersectL(B)
    =Aclosure intersect B intersect L(B)
    I'm not entirely sure if this is correct
    c) I'm not sure how to get started
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  2. #2
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    In part (a) it says union.
    In your attempt you changed it to intersection. WHY?
    What is the definition of limit point in your text material?
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  3. #3
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    I changed it to intersection because these were the original problems. My instructor asked us to consider these problems with the idea of intersection
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  4. #4
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    Quote Originally Posted by kathrynmath View Post
    I changed it to intersection because these were the original problems. My instructor asked us to consider these problems with the idea of intersection
    You did not answer the question about definitions about limit points.

    In the case of intersection there is absolutely nothing to prove.

    Only in the case of union is there any interesting about the problem.
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  5. #5
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    We have a limit point x of a set A if every neighborhood V(x) of x intersects the set A in some point other than x
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  6. #6
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    so that shows a)
    For b) I'm not quite sure if what I did was correct
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