I need to show that $(a,b)\times (c,d)$ is an open set in $\mathbb{R}^2$ with the Euclidian metric. I know that a set $U$ is open if for $x\in U$ there exists an open ball $B_\epsilon(x)$ such that $B_\epsilon(x)\subset U$ for some $\epsilon >0$. The euclidian metric in $\mathbb{R}^2$ is given by $d_2(x,y) = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2}$, so I need to show that for $x\in (a,b)\times (c,d)$ there exists $B_\epsilon(x) = \{y\in\mathbb{R}^2:\sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2} < \epsilon\}$ such that $B_\epsilon(x)\subset (a,b)\times (c,d)$ for some $\epsilon >0$, but I have no idea how!
Question: How do I show that $(a,b)\times (c,d)$ is an open set in $\mathbb{R}^2$ with the Euclidian metric?
$x=(x_1,x_2)$ 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?