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Math Help - Prove that a set is open.

  1. #1
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    Prove that a set is open.

    I need to prove that the following sets (in the complex plane) are open:

    1) |z-1-i|>1

    2) |z+i| =/= |z-i|

    I have a proof in my textbook for |z|<1 is open, using an epsilon and the triangle inequality, and I know that I need to do a similar thing for 1) here, but I can't see how to adapt the proof. I'm not really sure about 2) at all.

    Any help would be greatly appreciated, thank you.

    (My definition for a set S being open, is that from any point z in S, there is room to move some fixed positive distance in any direction without straying outside S.)
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  2. #2
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    Quote Originally Posted by charlie View Post
    I need to prove that the following sets (in the complex plane) are open:

    1) |z-1-i|>1

    2) |z+i| =/= |z-i|

    I have a proof in my textbook for |z|<1 is open, using an epsilon and the triangle inequality, and I know that I need to do a similar thing for 1) here, but I can't see how to adapt the proof. I'm not really sure about 2) at all.

    Any help would be greatly appreciated, thank you.

    (My definition for a set S being open, is that from any point z in S, there is room to move some fixed positive distance in any direction without straying outside S.)

    write |z-1-i|>1 as |z-(1+i)|>1. Everything that is shaded would be an open ball and everything outside of it. So it is certainly open. I will think about how to do an epsilon delta proof.
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  3. #3
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    Thank you! I can see that they are open, it's just the actual proofs that I'm having trouble with
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