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."

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- Feb 25th 2009, 11:02 AMHenryBConnected subsets of {(z,w) in C^2: z not equal to w}?
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." - Feb 26th 2009, 11:05 AMLaurent
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.