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Topology Proof?

  1. Oct 7, 2007 #1
    How can you prove sets
    1---------
    how can u prove the following sets are are open,
    a. the left half place {z: Re z > 0 };
    b. the open disk D(z0,r) for any [math]z_0 \varepsilon C[/math] and r > 0.

    2---------
    a. how can u prove the following set is a closed set:
    _
    D(z0, r)


    MY WORKING SO FAR
    1.. could you please give me a hint on how to start a and b as ive researched but still havent got much of an idea. once i get a little hint then ill try solving and show you my working..

    2a.
    --------
    if D(z0,r) is closed, this implies C\S (the compliment) is open. Therefore, for any z not belonging to the set, there is an e > 0 such that D(z,e) C C\S. This further implies z is not a limit point of S which means that it is a closed set?

    is this correct proof for 2a??
     
  2. jcsd
  3. Oct 7, 2007 #2

    HallsofIvy

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    Use the definition of open set, of course. If the real part of z is negative, can you find a disk about z such that every point in it has real part negative? (b) is a little harder. Show that for any point p in D(z0,r) there exist a disk about p that is a subset of D(z0,r). You will need the triangle inequality.

    No, you certainly cannot start a proof That [itex]\overline{D(z0,r)}[/itex] is closed by saying "if [itex]\overline{D(z0,r)}[/itex] is closed"!
    What, exactly, is your definition of [itex]\overline{D(z0,r)}[/itex]?
     
  4. Oct 7, 2007 #3
    well my definition of [itex]\overline{D}(z0,r)}[/itex] was that it is a set which is a closed disk??

    [itex]\overline{D(z0,r)}[/itex] { w: |z0 - w | < r }
     
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