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Math Help - Connected subsets of {(z,w) in C^2: z not equal to w}?

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    Connected subsets of {(z,w) in C^2: z not equal to w}?

    Find the connected components of X = \{(z,w) \in C^2 : z \neq w \} with the topology induced from C^2.

    The trouble with his is that I can't visualise the space X. I tried the same question with \Re instead of C and got the sets \{(x,y) \in \Re^2 : x > y \} and \{(x,y) \in \Re^2 : x < y \}, but can't see how to do the "complex version."
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  2. #2
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    Quote Originally Posted by HenryB View Post
    Find the connected components of X = \{(z,w) \in C^2 : z \neq w \} with the topology induced from C^2.

    The trouble with his is that I can't visualise the space X. I tried the same question with \Re instead of C and got the sets \{(x,y) \in \Re^2 : x > y \} and \{(x,y) \in \Re^2 : x < y \}, but can't see how to do the "complex version."
    The set X is connected.

    This is obvious if you think the following way: Let (x,y),(z,w)\in X (imagine four points on the complex plane). They are path-connected in X if there is a path \gamma_1:[0,1]\to\mathbb{C} from x to z and a path \gamma_2:[0,1]\to\mathbb{C} from y to w such that, for every t\in[0,1], \gamma_1(t)\neq \gamma_2(t) (this assures that the path (\gamma_1,\gamma_2) keeps inside X). There are plenty of such paths... On \mathbb{R}, if x<y and z>w, the paths had to meet for some t\in(0,1) due to the intermediate value theorem.
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