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?
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.