Problem regarding the equivalence relation

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Show that for some topological spaces (X, S) and (Y, T ), there is a closed set D in X × Y with product topology which is not in any product σ-algebra A ⊗ B, for example if A and B are the Borel σ-algebras for the given topologies. Hint: Let X = Y be a set with cardinality greater than c, for example, the set 2I of all subsets of I := [0, 1] (Theorem 1.4.2). Let D be the diagonal {(x, x ): x ∈ X }. Show that for each C ∈ A ⊗ B, there are sequences { An } ⊂ A and {Bn } ⊂ B such that C is in the σ-algebra generated by { An × Bn }n ≥ 1. For each n, let x =n u mean that x ∈ An if and only if u ∈ An . Define a relation x ≡ u iff for all n, x =n u. Show that this is an equivalence relation which has at most c different equivalence classes, and for any x, y, and u, if x ≡ u, then (x, y) ∈ C if and only if (u, y) ∈ C . For C = D and y = x , find a contradiction.

Reference no: EM13966754

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