Find the connected components of with the topology induced from .
The trouble with his is that I can't visualise the space X. I tried the same question with instead of and got the sets and , but can't see how to do the "complex version."
Find the connected components of with the topology induced from .
The trouble with his is that I can't visualise the space X. I tried the same question with instead of and got the sets and , but can't see how to do the "complex version."
The set is connected.
This is obvious if you think the following way: Let (imagine four points on the complex plane). They are path-connected in if there is a path from to and a path from to such that, for every , (this assures that the path keeps inside ). There are plenty of such paths... On , if and , the paths had to meet for some due to the intermediate value theorem.