3
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Which of the following sets are dense in $\Bbb R^2$ with respect to the usual topology.

  1. $\{(x,y) \in\mathbb{R}^2:x\in \mathbb{N}\}$
  2. $\{(x,y) \in\mathbb{R}^2:x+y \text{ is a rational number}\}$
  3. $\{(x,y) \in\mathbb{R}^2:x^2+y^2=5\}$
  4. $\{(x,y) \in\mathbb{R}^2:xy\neq 0\}$

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 .

  • 0
    Looks good to me.2012-12-17
  • 2
    I assume that the $\mathbb{R}$ in the title is intended to be $\mathbb{R}^2$.2012-12-17

2 Answers 2