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?

If you can use continuous functions, then the set in question is the inverse image of the open interval $(0,\infty)$ under the continuous function $\mathbb R^2 \to \mathbb R$ with $(x,y)\mapsto xy$.
