What is the Maximum possible number of connected components of $ \Bbb R ^3 \setminus (H_1\cup H_2 \cup H_3 \cup H_4)$, where $H_i (i=1,2,3,4)$ are hyperplanes in $\Bbb R^3$?
Intuitively I calculated that If there is only $1$ Hyperplane then the maximum number of connected components is $2$, if there is $2$ Hyperplanes then the number of connected components is $4$, and for $3$ the number is $8$. But for $4$ of them, I cant imagine what it would be like. Since , for the previous $3$ cases the number is $2^n$ , where $n=1,2,3$, I think the answer will be $2^4$. But I cant find any rigorous proof. Please help.