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Borel family

  1. Jan 20, 2008 #1
    Can anybody suggest how to write an open interval (a,b) as a combination(union, intersection and compliment) of closed intervals of the form [c,d] and vice versa.
    What if closed intervals are half closed as following (-inf, f]. 'f' being rational.
     
  2. jcsd
  3. Jan 21, 2008 #2

    CompuChip

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    What about something like

    [tex](a, b)^C = (-\infty, a] \cup [b, \infty) [/tex]
     
  4. Jan 24, 2008 #3
    Probably you mean not a finite combination, but the union of an infinite sequence, like
    [tex](a,b) = [a+1,b-1] \cup [a-0.5,b+0.5] \cup\dots[/tex]
     
  5. Jan 25, 2008 #4
    I think both of them are right. I was initially confused whether to consider (-inf,a] as closed set or not.
    Thanks.
     
  6. Jan 25, 2008 #5

    CompuChip

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    It's not, and it's not open either. But I kind of hoped you would see how to write (-inf, a] as a union of closed sets. And I don't think a finite combination is possible, since any finite union or intersection of closed sets is closed, right?
     
  7. Jan 25, 2008 #6
    Intersection, not union here. Assuming the first one on the right side was supposed to be [a-1,b+1] then this union is equal to [a-1,b+1].
     
  8. Jan 25, 2008 #7
    It should be closed, as it is the complement of an open set (a, inf) which is open.
     
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