# Product metric space

• Mar 16th 2011, 10:59 PM
jackie
Product metric space
I am trying to show that if U is open in a metric space $(X,d)$ and $V$ is open in $(Y,p)$. Then $U \times V$ is open in the product topology generated by the taxicab metric $(X \times Y,e)$. In order to show this, I tried to prove that $B_e((x,y),\epsilon)=B_d(x,\epsilon) \times B_p(y,\epsilon)$.
Let $(a,b)\in B_e((x,y),\epsilon)
\Leftrightarrow e((a,b),(x,y)) < \epsilon
\Leftrightarrow d(a,x)+p(b,y) < \epsilon
\Leftrightarrow d(x,a) < \epsilon$
and $p(b,y)< \epsilon
\Leftrightarrow a\in B_d(x,\epsilon)$
and $y\in B_p(y,\epsilon)$

However, there is a problem with the 3rd equivalent inequality since it only holds one direction, not the other. I am not sure if I approach this problem correctly. Hope someone can help me on this.
• Mar 17th 2011, 12:21 AM
FernandoRevilla
You only need to prove that for every $(a,b)\in U\times V$ there exists an open ball $B_e((a,b),\epsilon)\subset U\times V$ and not necessarily that $B_e((a,b),\epsilon)\subset U\times V$ is the product of two open balls.