Is it possible to construct a family of sets $\{A\_{ij}\}\_{i,j=1}^\infty$ and $\{B\_{ij}\}\_{i,j=1}^\infty$ such that:


$(1)$For any positive integer $i\geq1$,$A\_{ij}\searrow\emptyset$ and $B\_{ij}\searrow\emptyset$ as $j\to\infty$;


$(2)\bigcup\_{i=1}^\infty A\_{i1}=\bigcup\_{i=1}^\infty B\_{i1}$;


$(3)A\_{11},A\_{21},A\_{31}...$are pairwise disjoint and $B\_{11},B\_{21},B\_{31}...$are pairwise disjoint;


$(4)$For any positive integers $i$,there exist positive integers $j\_1,j\_2$ and sequences of positive integers $\{i\_k\}\_{k=1}^\infty,\{j\_k\}\_{k=1}^\infty$ such that $A\_{ij\_1}\bigcap (\bigcup\_{k=1}^\infty B\_{ki\_k})=\emptyset$ and $B\_{ij\_2}\bigcap (\bigcup \_{k=1}^\infty A\_{kj\_k})=\emptyset$;


$(5)$For any sequences of positive integers $\{i\_k\}\_{k=1}^\infty$ and $\{j\_k\}\_{k=1}^\infty$,
$(\bigcup\_{k=1}^\infty A\_{ki\_k})\bigcap (\bigcup\_{k=1}^\infty B\_{kj\_k})\neq\emptyset$.



Construction of a family of sets satisfying specific properties
I found the contruction satisfying the properties above do not exist.Suppose for any positive integer $k$,the positive integers $i\_k$,$j\_k$ satisfy there exist sequences of positive integers $\{i\_{k,s}\}$ and $\{j\_{k,s}\}$ such that $A\_{k,i\_k}\bigcap (\bigcup\_{s=1}^\infty B\_{s,i\_{k,s}})=\emptyset$ and $B\_{kj\_k}\bigcap (\bigcup \_{s=1}^\infty A\_{sj\_{k,s}})=\emptyset$.
Then consider $\{C\_{k,i}=A\_{k,i\_k+i-1}\}\_{k,i=1}^\infty$ and $\{D\_{k,i}=B\_{k,j\_k+i-1}\}\_{k,i=1}^\infty$,they satisfy:


$(1)$For any positive integer $k\geq1$,$C\_{k,i}\searrow\emptyset$ and $D\_{k,i}\searrow\emptyset$ as $j\to\infty$;


$(2)$For any positive integers $k,j$,there exist positive integers $i\_1,i\_2$ such that $C\_{k,1}\bigcap D\_{j,i\_1}=\emptyset$ and $D\_{k,1}\bigcap C\_{j,i\_2}=\emptyset$.


Then use Ramiro de la Vega's answer for my earlier version of the question and we can get it is impossible to construct the families of sets.


