Cartesian product set difference I know how to handle the 2d case: http://www.proofwiki.org/wiki/Set_Difference_of_Cartesian_Products
But I am having trouble simplifying the following:
Let $X=\prod_{1}^\infty X_i, A_i \subset X_i$
How can I simplify/rewrite $X - (A_1 \times A_2 \times \cdots A_n \times X_{n+1} \times X_{n+2} \cdots)$ with unions/intersections?

You cannot describe the indicated set $S$ without using some negation. So let $A_j':=X_j \setminus A_j$ and $\hat A_j:=\pi_j^{-1}(A_j')=\lbrace x=(x_i)_{i\geq1}\in X | x_j\in A_j'\rbrace$. Then the set $S$ can be written as $S=\bigcup_{1\leq j\leq n} \hat A_j$, because an $x\in X$ is a member of $S$ iff at least one of the first $n$ "coordinates" $x_j$ of $x$ does not belong to the corresponding $A_j$.
