Let $\mathbb{Q}$ be the set of rational numbers endowed with a topology (let's say subspace topology). What could we say about its fundamental $\pi_{1}$ and homology groups?
Presumably, $\pi_{0}(\mathbb{Q}) = \mathbb{Q}$ since rationals are totally disconnected. I would also think that $\pi_{1}(\mathbb{Q},x_{0}) = \mathbb{Q}$ for some $x_{0} \in \mathbb{Q}$ since no two points of rationals can be connected by a loop. What about the homology groups?
Could anyone confirm? Or does anyone have an idea on how to proceed? Or maybe some relevant references?
EDIT: We need of course a basepoint for a fundamental group of space that is not path-connected and applying Hurewicz was a very poor choice. Thanks for correcting.