**Question:** Show that $\pi_{1}({\mathbb R}^{2} - {\mathbb Q}^{2})$ is uncountable.

**Motivation:** This is one of those problems that I saw in Hatcher and felt I should be able to do, but couldn't quite get there.

**What I Can Do:** There are proofs of this being path connected (though, I'm not exactly in love with any of those proofs) and this tells us we can let any point be our base-point. Now, let $p$ be some point in ${\mathbb R}^{2} - {\mathbb Q}^{2}$ and let's let this be our base-point. We can take one path from $p$ to $q$ and a second one from $q$ to $p$, and it's not hard to show that if these paths are different then there is at least one rational on the "inside" of it. Since there are uncountably many $q$, this would seem to imply uncountably many different elements of the fundamental group; the problem I'm having is showing that two loops like we've described are actually different! For example, a loop starting at $p$ and passing through $q$ should be different from a loop starting at $p$ and passing through $q'$ for none of these points the same, for at least an uncountable number of elements $q'$. Is there some construction I should be using to show these elements of the fundamental group are different?