# Is there a space $X$ such that $S^1$ is homeomorphic to $X \times X$? [duplicate]

Is there a space $X$ such that $S^1$ is homeomorphic to $X \times X$?

If there is it would need to have fundamental group $\mathbb{Z}$.

$\pi_1(X \times X)=\pi_1(X) \times \pi_1(X)$

What can I do now?

• Observe that $\mathbb{Z}$ cannot be $G \times G$ for any $G$. This is quite easy. Apr 2 '17 at 22:27

Is there a group $G$ such that $\mathbf Z \cong G \times G$?
• @Walter When is $G \times G$ cyclic? Apr 2 '17 at 22:29