I have the following metric:
$d((a,b),(c,d))=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}$ or $0$ if $(a,b)=(c,d)$
the question is to show $R=\mathbb R^2\setminus\{(0,0)\}$ is disconnected with respect to $d$.
ive got 2 sets
$U=\{(x,y)\mid y<0\}$
$V=\{(x,y)\mid y\geq0\}\setminus\{(0,0)\}$
so i have $U\cup V=R$ so i just need to show $U$ and $V$ are open w.r.t the metric d
im having trouble finding open balls around points on the x-axis,say $(b,0)$. it seems to be the case if i need a ball of radius $\epsilon>0$ i only have the point $(b,0)$ when $\epsilon<b$. is this correct so i can say my sets are both open?
or maybe its easier to show they are both closed?
thanks