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Math Help - Open and connected sets

  1. #1
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    Open and connected sets

    The question:
    Is the region |z - 1 + i| \ge 2:
    a) Open?
    b) Connected?

    I always have trouble with these questions, since I'm not entirely sure what 'open' and 'connected' sets entail. My solution is a) no, b) no.

    My reasoning is that it's not open because the set is a closed circle in the complex plane, with the set lying outside this circle. My reasoning for it not being connected is simply because there's a hole in the set (I'm sure my reasoning here is insufficient).

    No solutions were provided for this question.

    Any assistance would be great!
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  2. #2
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    This set consists of all points on and exterior to the circle with center 1-i and radius 2.

    It's not open because it contains boundary points.

    It is connected. Any 2 points in the region can be connected via a polygonal path (a finite number of line segments connected end to end).
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  3. #3
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    Quote Originally Posted by Glitch View Post
    The question:
    Is the region |z - 1 + i| \ge 2:
    a) Open?
    b) Connected?
    My solution is a) no, b) no.
    A set is connected in the complex plane if and only if any two points in the set can be joined by an arc contained entirely within the set. Is that true of the given set? Put it this way, are there two points in the set such we must leave the set if going from one to the other along a continuous curve?
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  4. #4
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    Thanks guys!
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