Is the following set, closed, open, or neither?
$$
\left\{(x, y) \in \mathbb{R}^{2}: x \geq 0, y \geq 0 \text { and } x y>1\right\}
$$
I looked at the complement and (possibly incorrectly) deduced that it was neither closed nor open. I believe this means that the set itself cannot be closed or open either.
Best Answer
It is an open set. Note that your set is equal to$$\bigl\{(x,y)\in\mathbb R^2\mid x+y>0\text{ and }xy>1\bigr\}.$$Therefore, it is equal to $F^{-1}\bigl((0,\infty)\times(1,\infty)\bigr)$, with $F(x,y)=(x+y,xy)$. Besides, $F$ is continuous.