Is There an Easier Way to Prove S is Disconnected?

[kimkibun]

Homework Statement

Let S={zεℂ: |z|<1 or |z-2|<1}. show that S is not connected.

Homework 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' $\subseteq B'$.

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

 

[haruspex]

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' $\subseteq B'$.

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' $\subseteq B'$?

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

Looks good to me.

 

[kimkibun]

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' $\subseteq B'$?

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

Looks good to me.

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

 

[Dick]

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

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

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.