I am trying to show that if U is open in a metric space $\displaystyle (X,d)$ and $\displaystyle V$ is open in $\displaystyle (Y,p)$. Then $\displaystyle U \times V$ is open in the product topology generated by the taxicab metric $\displaystyle (X \times Y,e)$. In order to show this, I tried to prove that $\displaystyle B_e((x,y),\epsilon)=B_d(x,\epsilon) \times B_p(y,\epsilon)$.
Let $\displaystyle (a,b)\in B_e((x,y),\epsilon) \Leftrightarrow e((a,b),(x,y)) < \epsilon \Leftrightarrow d(a,x)+p(b,y) < \epsilon \Leftrightarrow d(x,a) < \epsilon$ and $\displaystyle p(b,y)< \epsilon \Leftrightarrow a\in B_d(x,\epsilon)$ and $\displaystyle y\in B_p(y,\epsilon)$
2. You only need to prove that for every $\displaystyle (a,b)\in U\times V$ there exists an open ball $\displaystyle B_e((a,b),\epsilon)\subset U\times V$ and not necessarily that $\displaystyle B_e((a,b),\epsilon)\subset U\times V$ is the product of two open balls.