Prove that $\{(x,y)\mid xy>0\}$ is open I need to prove this using open balls. So the general idea is to construct a open ball around a point of the set. A point $(x,y)$ such that $xy>0$. Then we must prove that this ball is inside the set. However, I don't know how to find a radius for this open ball. Can somebody help me in this proof?

Draw the set in question. It has an easy-to-describe boundary, $B$; given any point $(x, y)$ in the set, it should visually be easy to find the distance from $(x, y)$ to $B$. Any radius smaller than this distance will work.
