Show that {(x,y) in R2 | xy<7} is open (closed ?) with respect to the l1 metric I really need your help.
I have to determine if {(x,y) in R2 | xy<7} is open or closed or both or neither, with respect to the Manhattan metric (or l1 metric or taxicab metric).
I have the feeling that it is open but I don’t manage to prove it.
I wanted to use the property : U open and f continuous <=> f-1(U) open, but I cannot prove that f is continuous with respect to the Manhattan metric.
Thank you very much

The topology on $\mathbb R^2$ induced by the Manhattan metric coincides with the topology on $\mathbb R^2$ induced by the usual Euclidean metric.
This follows directly from the inequalities $d_E(x,y)\leq d_M(x,y)$ and $d_M(x,y)\leq2d_E(x,y)$.
So in this context $\mathbb R^2$ is equipped with its usual topology.
Further the function $f:\mathbb R^2\to\mathbb R$ prescribed by $\langle x,y\rangle\mapsto xy$ is continuous and the set $(-\infty,7)\subseteq\mathbb R$ is open.
We conclude that $f^{-1}((-\infty,7))=\{\langle x,y\rangle\mid xy<7\}$ is open.
The fact that $\mathbb R^2$ is connected then excludes that the set is also closed.
