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2nalfq8 2022-07-05 Answered
Suppose ( R , τ ) is the standard topological space. And B is the Borel σ-algebra from this space.
Define set A as:
A = { x R : x = q 1 n 1 + q 2 n 2  for some  q 1 , q 2 Q ,  and  n 1 , n 2 N }
How can I show that the set A belongs to B ?
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Answers (1)

Miguidi4y
Answered 2022-07-06 Author has 13 answers
By definition, every σ-algebra (in particular B ) is closed under countable union. Moreover, every singleton set { x } R belongs to B , since { x } is a closed set and a closed set is Borel. Consequently
A = q 1 , q 2 Q ,   n 1 , n 2 N { q 1 n 1 + q 2 n 2 }
is a countable union of sets in B , and thus also belongs to B .

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