Borel family

  • Thread starter gradnu
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  • #1
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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.
 

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  • #2
CompuChip
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What about something like

[tex](a, b)^C = (-\infty, a] \cup [b, \infty) [/tex]
 
  • #3
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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]
 
  • #4
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I think both of them are right. I was initially confused whether to consider (-inf,a] as closed set or not.
Thanks.
 
  • #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?
 
  • #6
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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]
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].
 
  • #7
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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?
It should be closed, as it is the complement of an open set (a, inf) which is open.
 

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