Thread: Prove that a set is open.

1. 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.)

2. Originally Posted by charlie
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 $\displaystyle |z-1-i|>1$ as $\displaystyle |z-(1+i)|>1$. 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.

3. Thank you! I can see that they are open, it's just the actual proofs that I'm having trouble with