# Is this complex set S a domain?

1. Jan 18, 2012

1. The problem statement, all variables and given/known data
Let S be the open unit disk of radius 2 centered at the origin. T is a subset of the real axis. The set R is obtained by removing T from S. Is R a domain when:
1. T is the line segment {z$\in$ ℂ | Re(z) ≤ 1 and Im(z) = 0}
2. T is the line segment {z$\in$ ℂ | Re(z) < 2 and Im(z) = 0}

2. Relevant equations
A set R is domain when it is open and connected.

3. The attempt at a solution
The problem is I've never taken a rigorous course on sets or proofs so I have very little knowledge in terms of how to see whether a set if open/connected.

1. It is a closed set because of the new boundary of S in the form of T.
2. It is open because Re(z) < 2 is included in the origin boundary for S.

2. Jan 18, 2012

### Dick

I don't think you need a rigorous course on sets or proofs. The question doesn't ask for a proof. But you do need at least a qualitative understanding of what 'open' and 'connected' mean. Your answers suggest you don't have that. What is your understanding of what 'open set' means? Put that together with a sketch of what those two regions look like.