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Conflicting statements from two topology textbooks

  1. Mar 11, 2008 #1
    Conflicting statements from topology textbooks

    Definitions: A point p is a limit point of A iff all open sets containing p intersects A-{p}. Let A' denote the set of all limit points of A. So far, so good.

    Cullen's topology book (1968) states that
    (A U B)' = A' U B'.
    I read her proof carefully and it looks good. Schaum's topology book (1965) also states the same thing and its proof also looks good.

    However, a university pdf solution to Munkres' problems (1st edition, p.100 #8d) states that (A U B)' = A' U B' is false. I read the discussion of the incorrect logic that is easily made in arriving at the "false conclusion." But Cullen's and Schaum's proofs don't use that incorrect logic, but they prove it differently.

    I'm so confused. Is it true or false?

    While we're at it. Is (A n B)' = A' n B' true? And what about infinite unions and infinite intersections? Oddly, I also haven't seen either statement in the form of a theorem (or a question asking for a proof) in any recent topology textbook.
     
    Last edited: Mar 11, 2008
  2. jcsd
  3. Mar 11, 2008 #2
    Ok, I found a counterexample to the infinite union case: A_n = (1/n, 1-1/n), n=1,2,3...., and to the infinite intersection case: A_n = (-1/n, 1/n), n=1,2,3....

    If the statement: A c B implies A' c B' is true (this is true, right?), then I have found that
    1) U(A') c (UA)'
    2) (A n B)' c A' n B' (I found a counterexample where equality does not hold)

    The finite union case gives equality IF (A U B)' = A' U B' is true.
     
    Last edited: Mar 12, 2008
  4. Mar 12, 2008 #3

    morphism

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    I'm pretty sure (A U B)' = A' U B' is true. Everything else you said is fine, too.
     
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