• Support PF! Buy your school textbooks, materials and every day products Here!

Closed set

  • #1

Homework Statement



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]

Homework Equations





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?
 

Answers and Replies

  • #2
354
0
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]​
 

Related Threads on Closed set

Replies
9
Views
2K
Replies
19
Views
8K
  • Last Post
Replies
8
Views
1K
Replies
2
Views
1K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
1
Views
617
  • Last Post
Replies
11
Views
2K
  • Last Post
Replies
12
Views
5K
  • Last Post
Replies
24
Views
6K
  • Last Post
Replies
1
Views
2K
Top