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.


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

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

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

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

*

*is not dense. The set of verticle lines with natural number $x$ coordinate is not dense. 

*This is dense. Given $(x,y)$, let $r = x + y$. If $r$ is irrational , let $q$ be any rational number close to $r$. Then $(x - (r - q), y)$ has rational sum and gets close to $(x,y)$. 

*This is a circle of radius of radius $\sqrt{5}$ which is not dense. 

*This is dense. You can get arbitrary close to any $(x,y)$ without intersecting the $x$ or the $y$ axis. 
