# Topology (complex analysis)

1. Feb 12, 2013

### kimkibun

1. The problem statement, all variables and given/known data
Let S={zεℂ: |z|<1 or |z-2|<1}. show that S is not connected.

2. Relevant equations
My prof use this definition of disconnected set.

Disconnected set - A set S $\subseteqℂ$ is disconnected if S is a union of two disjoint sets A and A' s.t. there exists open sets B and B' with A $\subseteq B$
and A' $\subseteq B'$.

3. The attempt at a solution

So here's my solution. I let A={zεℂ: |z|<1} and A'={zεℂ: |z-2|<1}. in order for me to show that S is disconnected set, i need to show the following.

i.) S=A U A'
ii.) A and A' are disjoint
iii.) A and A' are open sets.

Am i doing the right way? Thanks

2. Feb 12, 2013

### haruspex

That doesn't work. is it perhaps: A set S $\subseteqℂ$ is disconnected if S is a union of two sets A and A' s.t. there exists disjoint open sets B and B' with A $\subseteq B$ and A' $\subseteq B'$?
Looks good to me.

3. Feb 12, 2013

### kimkibun

sorry, you're right, i forgot the "disjoint" word.

is there any other way (probably, easier) to prove that S is a disconnected set?

4. Feb 12, 2013

### Dick

You are doing fine. You just have the three (easy!) things to prove about A and A'. And you may not have to do a terribly formal proof of each. Just indicate why they are true.

Last edited: Feb 12, 2013