I know that $\mathbb{R}^2$ with the post office metric is not separable. And the post office metric is defined by $m(x,y) = d(x,0) + d(y,0)$ for distinct points $x$ and $y$, and $m(x,x) = 0$ where $d$ is metric on $\mathbb{R}^2$.
My idea for the proof: For every $x \in \mathbb R^2$ we can find an $r_x$ such that $x$ is seperated from other elements of $\mathbb R^2$ (for instance if $r_x = d(x,0)$). And for a $D$ which is dense in $\mathbb R^2$ we can find a $y \in D$ st $y \in B(x,r_x)$. Then, we can match each $x \in \mathbb R^2$ with a $y \in D$. And since $\mathbb R^2$ is uncountable, $D$ is also uncountable. Thus, $\mathbb R^2$ is not separable since any dense set in it is uncountable.
Is my proof correct? I am not sure since $r_x$ depends on $x$.