Which of the following sets are dense in $\Bbb R^2$ with respect to the usual topology.
- $\{(x,y) \in\mathbb{R}^2:x\in \mathbb{N}\}$
- $\{(x,y) \in\mathbb{R}^2:x+y \text{ is a rational number}\}$
- $\{(x,y) \in\mathbb{R}^2:x^2+y^2=5\}$
- $\{(x,y) \in\mathbb{R}^2:xy\neq 0\}$
Clearly 1 is false.
3 is false as it is bounded and closed
4 is true as it is the set of all points that are not on the axes x and y.
Am I correct.
But I am not sure about 2 but my guess is true as rationals/ irrationals are dense and 2 holds iff either both are rational or conjugate irrational .