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Closed and Open Sets in R^d

  1. Sep 5, 2011 #1
    This is the question:
    Let A be an open set and B a closed set. If B ⊂ A, prove that A \ B is
    open. If A ⊂ B, prove that B \ A is closed.


    Right before this we have a theorem stated as below:
    In R^d,
    (a) the union of an arbitrary collection of open sets is open;
    (b) the intersection of any finite collection of open sets is open;
    (c) the intersection of an arbitrary collection of closed sets is closed;
    (d) the union of any finite collection of closed sets is closed.

    So in each case, I think we can just say A\B=A∩B^c (B^c means B complement), and since both A and B^c are open (by assumption and then by definition of a closed set where if B is closed, B-complement is open) we use part (b) of the previous theorem, A\B is open. But in this proof, I didn't use the assumption that B⊂A, so I know it isn't right. Same goes for the second part of the question. PLEASE HELP!!!!!
     
  2. jcsd
  3. Sep 5, 2011 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    By what logic do you "know it isn't right"? Pehaps this particular part does NOT depend on B being a subset of A. For example, if A and B were disjoint, A\B= A which is, just as your proof would assert, open. The fact is that, if A is open and B is closed, the A\B is open and B\A is closed, whether A is a subset of B or B is a subset of A or not.

     
  4. Sep 5, 2011 #3
    Ah, I see. Thanks, this makes a lot of sense now! I guess all I was thinking is that if in the question they indicated the "if"s I've got to be doing something wrong! hah! Thanks a lot for your help!!
     
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