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 $\mathbb{R}^2$?
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.
This sounds wrong. Maybe I can't see the picture, but if you take $A={0}$ and $B={0,1}$, then the first space will have two connected components and the second three.