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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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  ?

Miguidi4y · Accepted Answer

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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