Writing Open Intervals as Closed Intervals (-inf,f]

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Discussion Overview

The discussion revolves around the representation of open intervals as combinations of closed intervals, specifically exploring how to express intervals of the form (a, b) and (-inf, f] using unions, intersections, and complements of closed intervals. The conversation also touches on the nature of closed and open sets in this context.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant asks for suggestions on expressing open intervals (a, b) as unions or intersections of closed intervals, including half-closed intervals like (-inf, f].
  • Another participant proposes that the complement of an open interval (a, b) can be expressed as (-inf, a] ∪ [b, ∞).
  • A different participant suggests that the representation may require an infinite sequence, indicating that (a, b) could be expressed as a union of intervals like [a+1, b-1] and [a-0.5, b+0.5].
  • One participant expresses confusion about whether (-inf, a] should be considered a closed set and acknowledges the complexity of the problem.
  • Another participant asserts that (-inf, a] cannot be expressed as a finite combination of closed sets, emphasizing that finite unions or intersections of closed sets remain closed.
  • There is a correction regarding the nature of the sets discussed, indicating that (-inf, a] should be viewed as the complement of an open set (a, inf), which is open.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best way to express open intervals as combinations of closed intervals. Multiple competing views and interpretations are presented, particularly regarding the nature of closed and open sets.

Contextual Notes

There are limitations in the discussion regarding assumptions about the nature of closed sets and the definitions used for open and closed intervals. The possibility of finite versus infinite combinations of intervals remains unresolved.

gradnu
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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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What about something like

(a, b)^C = (-\infty, a] \cup [b, \infty)
 
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
 
I think both of them are right. I was initially confused whether to consider (-inf,a] as closed set or not.
Thanks.
 
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?
 
tsirel said:
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

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].
 
CompuChip said:
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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