Is the following set path connected?

Question

Let $A$ be the set of all those points of plane $\mathbb{R} ^2$ in which both coordinates are rational or both are irrational. Is $A$ path connected?

Answer 1

I assume that you mean that $A=\Bbb Q^2\cup\Bbb P^2$, where $\Bbb Q$ is the set of rational numbers, and $\Bbb P$ is the set of irrational numbers. (In other words, either both coordinates are rational, or both are irrational.)

HINT: If $L$ is a line of slope $1$ or $-1$, and $L\cap\Bbb Q^2\ne\varnothing$, then $L\subseteq A$. This gets you path-connectedness of $\Bbb Q^2$; for path-connectedness of $A$ you have to work a bit harder. For this you can use the suggestion in the extra credit paragraph of the answer to the earlier question in favor of which this was closed. You'll want to use the fact that if $a<b$ and $c<d$, there is an order-isomorphism between $(a,b)\cap\Bbb Q$ and $(c,d)\cap\Bbb Q$.