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Digital topology

  • Thread starter cragar
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  • #1
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Homework Statement


Show that the set of odd integers is dense in
the digital line topology on [itex] \mathbb{Z} [/itex]

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.
 

Answers and Replies

  • #2
Dick
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Can you spell out what the 'digital topology' is? {m} doesn't mean much to me.
 
  • #3
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I think I got it figured out. thanks for having me define my terms better.
 
  • #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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