Assume that $A,B \subset R^2$ and $A,B$ are compact and connected. And $A,B$ are homeomorphic. are$A^c$ and $B^c$ homeomorphic ?
I don't have a good idea to solve that.
Asked
Active
Viewed 235 times
0
-
2More interesting: if $A$ and $B$ are compact, connected and homeomorphic. Are the complementary sets homeomorphic? – Emanuele Paolini Dec 04 '15 at 13:25
1 Answers
3
Hint. Think of a set $A$ which is connected, but not simply connected, i. e. it has "holes", for example think of the unit circle $A = S^1 \subseteq \mathbf R^2$. Then $A^c$ is not connected.
martini
- 84,101
-
-
Added something. If only one of $A$, $B$ is simply connected, $A^c$ and $B^c$ are not homeomorphic. – martini Dec 04 '15 at 13:23
-
What if $A$ and $B$ be homeomorphic ? I think this is a better question. – mahdi mz Dec 04 '15 at 13:27
-
Then it is a duplicate: http://math.stackexchange.com/questions/1354119/are-the-complements-of-two-homeomorphic-compact-connected-subsets-of-mathbbr – martini Dec 04 '15 at 13:29