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Notation on Disks

  1. Jan 25, 2014 #1
    Hello,

    I'm reading "Complex Made Simple" by David Ullrich.


    He has these notation for disks

    [itex] D(z_0,r) = \left\{ z \in \mathbb{C}: |z-z_0|< r \right\} [/itex]

    [itex] \bar{D}(z_0,r) = \left\{ z \in \mathbb{C} : |z - z_0| \leq r \right\}[/itex]

    I understand that these sets are to be the open and closed disks with radius r respectively.

    The one I'm not sure about is what does [itex] \overline{D(z_0,r)} [/itex] mean? Any thoughts?
     
  2. jcsd
  3. Jan 25, 2014 #2

    Office_Shredder

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    That means the topological closure of the set [itex] D(z_0, r) [/itex]. It turns out to be equal to [itex] \overline{D}(z_0, r)[/itex] but they probably plan on proving that at some point.
     
  4. Jan 25, 2014 #3
    Oh thanks so much! This book doesn't assume topology, but one thing I've always been confused on is that

    if

    [itex] \overline{D(z_0,r)} = D(z_0,r) \cup \bar{D}(z_0,r) [/itex],



    why change the notation? I see you said that they turn out to be equal. Is this to specify a more theoretical idea than a practical idea?
     
  5. Jan 27, 2014 #4

    WWGD

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    Saying it a bit differently from O.Shredder, it is not immediate that what is called (kind of confusingly) a closed ball--your definition in the bottom --is not a closed set, and, like Office Shredder said, this will be proved at some later point in the book.
     
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