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Let $\tau \in \mathbb{S}^1$ be such that $\tau$ is not a root of unity. Let $E_\tau$ be the quotient space $S^1/\tau^{\mathbb Z}$. Consider it as a pointed space with basepoint the equivalence class of $1$. This is a path-connected space.

What is the fundamental group of this space?

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    @George: the Kronecker foliation is not the quotient space of the torus by that subgroup, it is *the foliation* by the cosets of that subgroup. The second is an abelian group with an useless topology; the first one is, well, a foliation.2010-12-28

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Can you see what the topology of the quotient $S^1/(\tau)$ is? Its open sets "are" the open sets in $S^1$ which are invariant under multiplication by $\tau$.

Once you see what the topology is, your question becomes easy.