tupules $(x,y)$ with at least one entry rational is connected in $R^2$

Question

I have studied connectedness and came across a problem which goes like this.. all the tuples $(x,y)$ with at least one entry rational is connected in $\Bbb R^2$. I have tried to prove it by contradiction but it didn't help

Answer 1

Hint: show that it is path connected by constructing a generic path in this subset from a generic point to $(0,0)$.

Another very useful question: is there an easy way to always reach one of the $x$- or $y$-axis of $\mathbb R^2$ without leaving the subset? (the $x$- and $y$-axis are of course in this set).

Answer 2

Let $\;(x,y)\in X:=\{\;(r,s)\in\Bbb R^2\;:\; r\in\Bbb Q\;\;or\;\;s\in\Bbb Q\;\}\;$, then if for example $\;x\in\Bbb Q\;$ and WLOG $\;x,y\ge 0\;$ , take a look at

$$\;C_1:=\{(x,b)\in\Bbb R^2\;:\;0\le b\le y\}\;,\;\;C_2:=\{(a,0)\in\Bbb R^2\;;\;0\le a\le x\}\;$$

Check that both $\;C_1,C_2\subset\in X\;$ and thus $\;C_1\cup C_2\;$ is a path from any element in $\;X\;$ to $\;(0,0)\;$ , making $\;X\;$ path-connected.

End the argument now.