Is it possible to prove closure of decidable languages under union and intersection, using enumerators?
Given a decidable language $L$ consider the following enumerator:

EnumL
 loop on strings x over 0 and 1 in canonical/lexicographic order
   if x in L then print x
 end
end


This procedure enumerates/emits all strings in $L$ in lexicographic order. So you can decide if $x \in L$ by running EnumL until it prints the first string of length $|x| + 1$. If it does print $x$ until that time then it means $x \in L$, otherwise $x \notin L$. Thus, using both enumerators EnumA and EnumB you can decide $x \in A\cup B$ and $x \in A\cap B$. In other words, if one of EnumA or EnumB eventually emits $x$ then $x \in A\cup B$, and if both EnumA and EnumB emit $x$ then $x \in A\cap B$.