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Digital topology

  1. Sep 11, 2012 #1
    1. The problem statement, all variables and given/known data
    Show that the set of odd integers is dense in
    the digital line topology on [itex] \mathbb{Z} [/itex]
    3. The attempt at a solution
    if m in Z is odd then it gets mapped to the set {m}=> open
    .
    So is the digital line topology just the integers.
    If I was given any 2 integers I could find an odd one in between if there is an element in between.
    If I was given to consecutive integers I wouldn't be able to find an odd one in between but there are no elements in between in this set. And I thinking about this question correctly.
     
  2. jcsd
  3. Sep 11, 2012 #2

    Dick

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    Can you spell out what the 'digital topology' is? {m} doesn't mean much to me.
     
  4. Sep 13, 2012 #3
    I think I got it figured out. thanks for having me define my terms better.
     
  5. Sep 13, 2012 #4

    SammyS

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    FYI:

    I Googled "digital line topology" and found the following:

    From http://www.math.csusb.edu/faculty/gllosent/About_me_files/555-Chapter2.pdf:
    Example 1.10.

    For each [itex]n\in\mathbb{Z}\,,[/itex] de fine:

    [itex]\displaystyle \textit{B}(n) =\left\{\matrix{\{ n\},\ & \text{if }\ n \text{ is odd.} \\ \ \\ \{ n-1,\,n,\,n+1\}, & \text{if }\ n \text{ is even.}}\right.[/itex]
    
    Consider [itex]\displaystyle {\frak{B}}= \{B(n)|n\in\mathbb{Z}\} [/itex]: a (basis of the*) digital line topology.​

    * added by me, SammyS.
     
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