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Homework Help: Is this complex set S a domain?

  1. Jan 18, 2012 #1
    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[itex]\in[/itex] ℂ | Re(z) ≤ 1 and Im(z) = 0}
    2. T is the line segment {z[itex]\in[/itex] 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. jcsd
  3. Jan 18, 2012 #2


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    Science Advisor
    Homework Helper

    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.
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