Finding a topology on $ \mathbb{R}^2 $ such that the $x$-axis is dense The problem is the following
Put a topology on $ \mathbb{R}^2$ with the property that the line $\{(x,0):x\in  \mathbb{R}\}$ is dense in $\mathbb{R}^2$
My attempt
If (a,b) is in $R^2$, then define an open sets of $(a,b)$ as the strip between $d$ and $c$, inclusive where $c$ has the opposite sign of $b$ and $d>b$ if $b$ is positive otherwise $d<b$ . Clearly this always contains the set $ \mathbb{R}\times\{0\}$. It also obeys the the topological laws, ie, the intersection of two open sets is open. The union of any number of open sets is open.
Thanks for your help

I think that you already get the solution. You also can put for open set the vertical strip  $| |$(assuming $\mathbb R^2$ a vertical strip)
