Every open set in R is a countable union of open intervals. Prove.

  • Context: Graduate 
  • Thread starter Thread starter seeker101
  • Start date Start date
  • Tags Tags
    intervals Set Union
Click For Summary
SUMMARY

The discussion centers on proving that every open set in R is a countable union of open intervals. Participants emphasize the necessity of incorporating rational numbers into the proof process. The conversation references Theorem 4.1.1 as a foundational element for various proof methods. Suggestions for alternative approaches are also solicited, indicating a collaborative effort to explore different proof techniques.

PREREQUISITES
  • Understanding of sigma algebras in measure theory
  • Familiarity with open sets and open intervals in real analysis
  • Knowledge of rational numbers and their properties
  • Experience with proof techniques in mathematical analysis
NEXT STEPS
  • Study the properties of sigma algebras generated by open sets and intervals
  • Explore Theorem 4.1.1 in detail for proof methodologies
  • Investigate the role of rational numbers in real analysis proofs
  • Learn about different proof techniques in topology and measure theory
USEFUL FOR

Mathematicians, students of real analysis, and anyone interested in the foundations of topology and measure theory.

seeker101
Messages
28
Reaction score
0
I was trying to prove that the sigma algebra generated by the set of open intervals is the same as the sigma algebra generated by the set of open sets. This proof devolves into proving the statement in the title. I think rational numbers must be brought into the picture to prove this stmt but I can't think of how to do it... Any suggestions?
 
Physics news on Phys.org

Similar threads

Undergrad Proof of a fact about real numbers - transcendental and algebraic
  • · Replies 7 ·
Replies
7
Views
2K
MHB Proof of a set union and intersection
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Collection of finite unions of half-open intervals form an algebra
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad The set of nonzero values of pmf is at most countable
  • · Replies 3 ·
Replies
3
Views
4K
MHB Proving Countability: Positive Rational Numbers & Union of Countable Sets
  • · Replies 11 ·
Replies
11
Views
3K
Undergrad Intersection & union of closed and open sets?
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Countably Infinite Unions and the Real Numbers: Can They Really Be Uncountable?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Dyadic cubes generate Borel sigma algebra of subset
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Thinking about equality of infinite sets
  • · Replies 5 ·
Replies
5
Views
3K
Graduate Can Open Subsets of Real Numbers Form Countable Unions of Intervals?
  • · Replies 1 ·
Replies
1
Views
2K