I wanted to calculate the fundamental group of $\mathbb{R}/\mathbb{Q}$ (with the quotient topology). I thought this could be fun because of the following proof idea:
$\mathbb{R}/\mathbb{Q}$ inherits a topological group structure from $\mathbb{R}$, so the fundamental group is abelian. Thus the fundamental group is abelian, and so the same as the first homology group. We can use the long exact sequence of homology at $n=1$ to get the exact sequence
$$0\to H_1(\mathbb{R}/\mathbb{Q})\to \mathbb{Z}^\mathbb{Q}\to\mathbb{Z}\to\mathbb{Z}\to 0$$
And then a bit of reasoning leads us to see $\pi_1(\mathbb{R}/\mathbb{Q})\cong H_1(\mathbb{R}/\mathbb{Q})\cong\mathbb{Z}^\mathbb{Q}$, this seemed like a reasonable conclusion so I was happy.
I have however thought some more about it and have found two problems in my proof. I am not a 100% sure that $\mathbb{R}/\mathbb{Q}$ really is a topological group. Is the fact that I am collapsing by a normal sub group enough to ensure that the action of $\mathbb{R}/\mathbb{Q}$ on itself and inversion are continuous?
The second problem is that $(\mathbb{R}, \mathbb{Q})$ is not a good pair, so using the long exact sequence is acutally a no go.
Do any of you kind folk know how to save my proof, or find a completely different method to calculate the fundamental group of $\mathbb{R}/\mathbb{Q}$?
Thanks in advance