If closed sets $A,B\subseteq\mathbb{R}^2$ are non-homeomorphic, can $\mathbb{R}^2\setminus A$ and $\mathbb{R}^2\setminus B$ be homeomorphic? I have a question. Could you please help me to solve this problem?
Is it possible that $\mathbb{R}^2\setminus A$ and $\mathbb{R}^2\setminus B$ are homeomorphic, when $A$ and $B$ are non-homeomorphic closed subsets of $\mathbb{R}^2$?

$A= \{(0,0)\}$, $B=B[0,1] $ i.e closed unit ball is an example of this.
