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Closed set

  1. Apr 29, 2010 #1
    1. The problem statement, all variables and given/known data

    on the complex line, with the usual metric, I need to determine if this is a closed set.

    [tex] A = \left\{\left|\frac{1}{z^{2}+1} \right|: |z| = 1 ; z\neq \pm i\right \} [/tex]

    2. Relevant equations

    3. The attempt at a solution

    A closed set implies that the set of all limit points belongs to A.

    Usually I'm given a function, and I take an arbitrary convergent sequence and show whether or not that point to which it converges is in A or not. But when I have just a set like this, I'm unsure of how to do that. Any advice?
  2. jcsd
  3. Apr 29, 2010 #2
    Try finding a way to figure out what points are in the set.

    Also, you might find it worthwhile to think of A as the image (your prof might call it the range) of the function [itex]f\colon \{z:|z|=1, z\neq\pm i\}\to\mathbb{C}[/itex] defined by
    [tex] f(z) = \left|\frac{1}{z^2+1}\right|.[/tex]​
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