Suppose that $E \subset \mathbb{C}$ is connected. Is it true that $\mathbb{C} \setminus E$ is also connected ?
The only few examples I have tell me it seems True but maybe there is a simple counter-example or a simple proof to this. Any help ?
Asked
Active
Viewed 44 times
0
-
1What if $E$ is the $x$-axis? – carmichael561 Apr 16 '18 at 15:39
-
Ooh right ! And if $E$ is bounded do we have the result ? – Atmos Apr 16 '18 at 15:39
-
1What is $E$ is a circle? – saulspatz Apr 16 '18 at 15:40
-
If $E$ is a circle, then $\mathbb{C} \backslash \left{E\right}$ is ( path )connected – Atmos Apr 16 '18 at 15:40
-
Atmos, not a ball, but think of $|z|=1$. – Clayton Apr 16 '18 at 15:45
-
Ah yes you are right, thank you – Atmos Apr 16 '18 at 15:45