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

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SUMMARY

This discussion focuses on expressing open intervals, specifically (a,b), as unions or intersections of closed intervals, such as [c,d]. Participants clarify that closed intervals can be represented as unions of infinite sequences rather than finite combinations. The discussion highlights that the complement of an open interval, such as (a, ∞), results in a closed interval, specifically (-∞, a]. The consensus is that finite unions or intersections of closed sets remain closed, and thus cannot represent open intervals.

PREREQUISITES
  • Understanding of interval notation, including open and closed intervals.
  • Familiarity with set theory concepts such as union, intersection, and complement.
  • Knowledge of rational numbers and their properties.
  • Basic understanding of infinite sequences and their implications in set theory.
NEXT STEPS
  • Research the properties of open and closed intervals in real analysis.
  • Study the concept of complements in set theory, particularly in relation to intervals.
  • Explore infinite unions and intersections in the context of topology.
  • Learn about the implications of rational numbers in interval notation and set representation.
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Mathematicians, students of real analysis, and anyone interested in advanced set theory and interval notation.

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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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