I'm interested in showing that there are no retractions $r:\mathbb{R}^3\rightarrow A$ where $A\subset \Bbb{R}^3$ such that $A$ is homeomorphic to $S^1$.
My idea is by contradiction. So when we assume that there exists $r$ as above then we can consider $$r_*:\Pi_1(\Bbb{R}^3)\rightarrow \Pi_1(A)$$ And since $r$ is a retract, $r_*$ is a surjection. Therefore we have a surjection between $\Pi_1(\Bbb{R}^3)$ and $\Pi_1(S^1)=\Bbb{Z}$. But now since $\Bbb{R}^3$ is starshaped $\Pi_1(\Bbb{R}^3)=\{e\}$, so trivial. But this gives us a contradiction.
Does this works?
Thanks a lot.