Fundamental group contains $\mathbb{R}$ or $\mathbb{Q}$

Question

Is there any topological space whose fundamental group contains $\mathbb{Q}$ or $\mathbb{R}$? In case of (singular) homology or cohomology, we can change its coefficients to any abelian groups (with suitable modification, such as universal coefficient theorem). But for a fundamental group, I've never seen any space that has fundamental group $\mathbb{Q}$ or $\mathbb{R}$. I think it is impossible, but how to prove it? Thanks in advance.

Answer 1

Any group (including $\mathbb{Q}$ and $\mathbb{R}$) can be realized as the fundamental group of some space - in fact, for every group $G$, there is a two-dimensional CW complex with fundamental group $G$. See Proposition 1.28 of Hatcher here.

Answer 2

Endow $Q$ with the discrete topology $Q_d$ and consider the classifying space corresponding to $Q_d$. Same construction with $R$.