Which of the following are Dense in $\mathbb{R}^2$? Which of the following sets are dense in $\mathbb R^2$ with respect to the usual topology.


*

*$\{ (x, y)\in\mathbb R^2 : x\in\mathbb N\}$

*$\{ (x, y)\in\mathbb R^2 : x+y\in\mathbb Q\}$

*$\{ (x, y)\in\mathbb R^2 : x^2 + y^2 = 5\}$

*$\{ (x, y)\in\mathbb R^2 : xy\neq 0\}$. 
Any hint is welcome. 

*

*No. It's a bunch of parallel lines.  These are vertical and the go through the integer points on the $x$-axis.

*No. It's a circle.

*Yes. It's the plane with the $x$ and $y$ axes excised.

*Interesting. It is a union of parallel lines with slope -1 and $y$-intercept at the various rationals. It's dense in the plane.
