1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Topology Proof?

  1. Oct 7, 2007 #1
    How can you prove sets
    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.

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

    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..

    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


    User Avatar
    Science Advisor

    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 }
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook