In Kirillov,Gvishiani, Theorems and Problems in Functional Analysis(Springer,1982) Exercise 71 (b) states
$(A_1\cap A_2)\Delta(B_1\cap B_2)\subset(A_1\Delta B_1)\cap(A_2\Delta B_2)$.
I tried with different methods, but I could prove only $(A_1\cap A_2)\Delta(B_1\cap B_2)\subset(A_1\Delta B_1)\cup(A_2\Delta B_2)$ (union instead of intersection in the right-hand-side). It can be a typo, but they remark that this inclusion means that the operation of intersection is continuous with respect to the "distance" $d(A,B):=A\Delta B$. I would appreciate for any proof or disproof. I don't understand this type of continuity.