Let $d$ be a metric on $\mathbb{R}^2$ defined as $d((x_1,y_1),(x_2,y_2))=\begin{cases} |y_1-y_2| \mbox{ if } x_1=x_2 \\ 1+|y_1-y_2| \mbox{ if } x_1 \neq x_2 \end{cases}$.
Let $N((x,y),\epsilon)$ be an open neighborhood in $(\mathbb{R}^2, d)$. If $0< \epsilon \leq 1$ then we have two cases. If $x_1 = x_2$, then we get a line segment between $0
I'm not sure if the above is correct. Nor am I sure of $\epsilon > 1$.