Let X = Y be uncountable and define A , B to be the countable-cocountable σ-algebras on X , Y respectively. Let C = σ ( A × B ). Prove that for each subset E ∈ C, there exist A , B in A , B countable such that either E ⊆ ( A × Y ) ∪ ( X × B ) or E c ⊆ ( A × Y ) ∪ ( X × B ).My idea is to define D as the collection of all such subsets of C, prove that D contains A 1 × B 1 for each A 1 ∈ A , B 1 ∈ B and then show that D is a σ-algebra.I am able to prove that D is a σ algebra, and I can show that in the case A 1 , B 1 are either both countable or cocountable, A 1 × B 1 is in D. How do I do it for the case that exactly one of them is cocountable?Something that may be useful is ( A 1 × B 1 ) c = ( A 1 c × Y ) ∪ ( X × B 1 c ) ∪ ( A 1 c × B 1 c )

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Let   X  =  Y be uncountable and define       A    ,      B   to be the countable-cocountable   σ-algebras on   X  ,  Y respectively. Let   C  =  σ  (  A  ×  B  ). Prove that for each subset   E  ∈  C, there exist   A  ,  B in       A    ,      B   countable such that either   E  ⊆  (  A  ×  Y  )  ∪  (  X  ×  B  ) or       E    c    ⊆  (  A  ×  Y  )  ∪  (  X  ×  B  ).My idea is to define   D as the collection of all such subsets of   C, prove that   D contains       A    1    ×  B  1 for each       A    1    ∈      A    ,      B    1    ∈      B   and then show that   D is a   σ-algebra.I am able to prove that   D is a   σ algebra, and I can show that in the case       A    1    ,      B    1   are either both countable or cocountable,       A    1    ×      B    1   is in   D. How do I do it for the case that exactly one of them is cocountable?Something that may be useful is  (      A    1    ×      B    1        )    c    =  (      A    1    c    ×  Y  )  ∪  (  X  ×      B    1    c    )  ∪  (      A    1    c    ×      B    1    c    )

Bruno Dixon · Accepted Answer

Suppose       A    1   is countable and       B    1   is cocountable. Then it follows that      A    1    ×      B    1    ⊆      A    1    ×  Y  ⊆  (      A    1    ×  Y  )  ∪  (  X  ×  B  )where   B  ∈      B   is an arbitrary countable set (e.g.,   B  =  ∅). This shows that       A    1    ×      B    1    ∈  D.

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