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Thread: product of closed maps

  1. #1
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    product of closed maps

    I have two closed maps:
    $f,g:\mathbb{C}^n\rightarrow \mathbb{C}.$

    I need to prove that their product:

    $fg:\mathbb{C}^n\rightarrow \mathbb{C}$

    is also a closed map.
    Is it an easy way to show that?
    Last edited by Plato; Jun 12th 2015 at 03:58 PM.
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  2. #2
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    Re: product of closed maps

    Hey Kaatie.

    Can you look at the intermediate stages? If you have two maps then you go A -> B -> C (when they are composed together) go from closed to closed to closed.

    How are you applying your maps if both go from C^n to C?
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  3. #3
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    Re: product of closed maps

    Is the map $\displaystyle \mathbb{C} \times \mathbb{C} \longrightarrow \mathbb{C}$ given by $\displaystyle (z_1, z_2) \mapsto z_1 z_2$ closed?

    Is the map $\displaystyle \mathbb{C}^n \longrightarrow \mathbb{C}^n \times \mathbb{C}^n $ given by $\displaystyle \zeta \mapsto (\zeta, \zeta)$ closed?

    By the way - once you get it, assuming you go this route, you should remember the general approach to this proof. It's a common technique in topology.
    Last edited by johnsomeone; Jun 12th 2015 at 11:39 PM.
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  4. #4
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    Re: product of closed maps

    Looking at my comment again, it's not clear to me that $\displaystyle \mathbb{C} \times \mathbb{C} \longrightarrow \mathbb{C}$ given by $\displaystyle (z_1, z_2) \mapsto z_1 z_2$ *is* a closed map.

    Maybe it is, but I don't see a quick & simple proof that is is, nor do I see a quick & simple counter-example. I'm not saying that they don't exist, just that I didn't find one after investing a few minutes.

    I'm not going to invest any more time, because you've recently posted a question about closed maps that was incorrect ( mathhelpforum.com/discrete-math/244690-prove-map-closed.html ). If your questions are speculations, please make that clear when you give them. Otherwise, you'll waste our time, because we'll be analyzing a situation based on a faulty assumption of the eventual correct answer.

    The approach chiro and I were suggesting is the standard way to prove propositions like yours. Maybe your proposition is true, and but this "standard approach" fails. Or maybe this standard approach works, and I just missed the easy proof. Or maybe your proposition itself isn't true. I don't know, and since I don't trust you, I'm not going to invest the time to figure it out.
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