# A set is open in two metric spaces?

For $(X,d_1)$ and $(X_2,d_2)$ is two metric space, set $(X=X_1 \times X_2)$ and $d(x,y)=max\{d_1(x_1,y_1),d_2(x_2,y_2)\}$ , $\overline{d}=d_1(x_1,y_1)+d_2(x_2,y_2)$ with $x=(x_1,x_2),y=(y_1,y_2)\in X$. Prove that a set $G$ is open in $(X,d)$ $\Longleftrightarrow$ $G$ is too open in $(X,\overline{d}).$ I try to use definition of open set: $G$ is open if $\forall x\in G: \exists B(x,r)\subset G$.

Other way, i think we can prove two metric is equivalent.( but i can't finish ). Can you help me. Thanks for your help.

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## migrated from mathoverflow.netFeb 24 '14 at 17:06

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Prove that $$d \le \bar d \le 2d.$$ Then notice that every ball in one matric contains a ball for the other metric, and vice-versa.