# Borel family

1. Jan 20, 2008

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. Jan 21, 2008

### CompuChip

$$(a, b)^C = (-\infty, a] \cup [b, \infty)$$

3. Jan 24, 2008

### tsirel

Probably you mean not a finite combination, but the union of an infinite sequence, like
$$(a,b) = [a+1,b-1] \cup [a-0.5,b+0.5] \cup\dots$$

4. Jan 25, 2008

I think both of them are right. I was initially confused whether to consider (-inf,a] as closed set or not.
Thanks.

5. Jan 25, 2008

### CompuChip

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. Jan 25, 2008

### d_leet

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. Jan 25, 2008

### d_leet

It should be closed, as it is the complement of an open set (a, inf) which is open.