Let X = Y be uncountable and define A , B to be the countable-cocountable...

Logan Wyatt

Logan Wyatt

Answered

2022-06-30

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 )

Answer & Explanation

Bruno Dixon

Bruno Dixon

Expert

2022-07-01Added 14 answers

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.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?