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