Cartesian product of open sets in metric spaces is open I am lost in trying to solve the following:
Let $(X_1,d_1)$ and $(X_2,d_2)$ be two metric spaces. Define
$$X = X_1 \times X_2 = \{ \mathbf{x}=(x_1,x_2) : x_1 \in X_1, x_2 \in X_2  \}$$
and $\forall \ \mathbf{x},\mathbf{y} \in X$
$$d_+(\mathbf{x},\mathbf{y})=d_1(x_1,y_1)+d_2(x_2,y_2) \quad d_e(\mathbf{x},\mathbf{y})= \sqrt{ (d_1(x_1,y_1)^2+d_2(x_2,y_2)^2) } \quad d_{\max}(\mathbf{x},\mathbf{y})= \max \{d_1(x_1,y_1), d_2(x_2,y_2)\}  $$
Show that:
If $U_1 \subset X_1$ is an open subset of $(X_1,d_1)$ and $U_2 \subset X_2$ is an open subset of $(X_2,d_2)$, then $U_1 \times U_2$ is an open subset of $(X,d_{+})$. What can we say about $(X,d_e)$ and $(X,d_{\max})$?
Thanks in advance!
 A: (I). We want to show that for any $(p_1,p_2)\in U_1\times U_2$ there exists $r_3>0$ such that  the  open $d_+$ ball $B_{d_+}((p_1,p_2),r_3)$ is a subset of $U_1\times U_2.$
Take $r_1,r_2>0$ such that  $B_{d_1}(p_1,r_1)\subset U_1$  and $B_{d_2}(p_2,r_2)\subset U_2.$ Let $r_3=\min (r_1,r_2).$  Then  $$(p,p')\in B_{d_+}((p_1,p_2),r_3)\implies$$ $$\implies (\;d_1(p,p_1)<r_3\leq r_1\;\land\; d_2(p',p_2)<r_3\leq r_2\;) \implies$$ $$\implies (\; p\in B_{d_1}(p_1,r_1)\subset U_1 \;\land \;p'\in B_{d_2}(p_2,r_2)\subset U_2\;)\implies$$ $$\implies (p,p')\in U_1\times U_2.$$
Since this holds for all $(p,p')\in B_{d_+}((p_1,p_2),r_3),$ we have $B_{d_+}((p_1,p_2),r_3)\subset U_1\times U_2. $ 
(II). Metrics $d$ and $d'$ on a set $X$ produce the same topology iff $$(i).\quad  \forall p\in X \;\forall r>0 \;\exists s>0 \;(\;B_d(p,s)\subset B_{d'}(p,r)\;),\text { and }$$ $$ (ii). \quad \forall p\in X \;\forall r>0 \;\exists s'>0\;(\;B_{d'}(p,s')\subset B_d(p,r)\;).$$ 
To get an idea of how to use this to show that $d_e$ and $d_{max}$ generate the same topology as $d_+,$ consider the case $X_1=X_2=\mathbb R,$ with $d_1(x,y)=d_2(x,y)=|x-y|.$  Sketch  some pictures of open balls of various radii, centered at some $p\in \mathbb R^2$, with respect to these 3 metrics.
Metrics on a set $X$ that generate the same topology  on $X$ are called equivalent metrics.
A: Hint:
It can be shown that the 3 mentioned metrics on $X_1\times X_2$ (even) induce the same topology. 
This on base of:


*

*$d_e(\mathbf{x},\mathbf{y})\leq d_+(\mathbf{x},\mathbf{y})$

*$d_+(\mathbf{x},\mathbf{y})\leq2d_{\text{max}}(\mathbf{x},\mathbf{y})$

*$d_{\text{max}}(\mathbf{x},\mathbf{y})\leq d_e(\mathbf{x},\mathbf{y})$

