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Math Help - How do you prove that z->z^2 is a closed map ?

  1. #1
    Aki
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    How do you prove that z->z^2 is a closed map ?

    I want to prove that the mapping of the complex plane to the complex plane z\mapsto z^2 is a closed map.
    Only way I can think of is to prove first that it is a proper map.
    Is there a more straightforward way ?
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  2. #2
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    Re: How do you prove that z->z^2 is a closed map ?

    It's obvious that (\cdot )^2 : \hat{\mathbb{C}}  \rightarrow \hat{\mathbb{C}} is closed, and since it leaves \infty fixed it behaves nice with respect to unbounded closed sets (an unbounded set is closed in the plane iff the set plus \infty is closed in the sphere). ( \hat{\mathbb{C}} denotes the Riemann sphere)
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  3. #3
    Aki
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    Re: How do you prove that z->z^2 is a closed map ?

    Thank you for your help.

    From your reply I got the impression that the necessary and sufficient condition for a continuous mapping of \mathbb{C} into \mathbb{C} to be a closed map is to leave \infty fixed.
    Am I right ?
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    Re: How do you prove that z->z^2 is a closed map ?

    Quote Originally Posted by Aki View Post
    Thank you for your help.

    From your reply I got the impression that the necessary and sufficient condition for a continuous mapping of \mathbb{C} into \mathbb{C} to be a closed map is to leave \infty fixed.
    Am I right ?
    Not at all: Take f,g: \mathbb{C} \rightarrow \mathbb{C} where f(x+iy)=x the natural projection on the real axis, then f fixes \infty (in the sense that there is a mapping of the sphere to the sphere that coincides with f via stereographic projection, ie. as usual), but it's not difficult to see that f is not closed. On the other hand if g is a constant, we can extend g to the sphere and this is, trivially, a closed continous map that doesn't fix \infty.

    What I meant to say was that obviously your mapping takes compact sets to compact sets (both in the plane), so we only need to check whether it sends unbounded closed sets (say F) to closed sets ( f(F) say), but by adding \infty to F we get a closed set G in the sphere and thus f(G) is closed in the sphere, but by the first post f(G)\setminus \infty is closed in the plane, but this is exactly f(F) and we're done.
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  5. #5
    Aki
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    Re: How do you prove that z->z^2 is a closed map ?

    Thank you for the detailed explanation.
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