Show that $(a,b)\times (c,d)$ is an open set in $\mathbb{R}^2$ with the Euclidian metric.
Solution 1:
$x=(x_1,x_2)$ with $a<x_1<b$ and $c<x_2<d$. Let $r=\min(x_1-a,b-x_1,x_2-c,d-x_2)$. This is the minimum distance from $(x_1,x_2)$ to the edges of the rectangle $(a,b)\times(c,d)$. Surely you can prove the open disc with centre $(x_1,x_2)$ and radius $r$ is contained within the open rectangle?