# Digital topology

1. Sep 11, 2012

### cragar

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 $\mathbb{Z}$
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. Sep 11, 2012

### Dick

Can you spell out what the 'digital topology' is? {m} doesn't mean much to me.

3. Sep 13, 2012

### cragar

I think I got it figured out. thanks for having me define my terms better.

4. Sep 13, 2012

### SammyS

Staff Emeritus
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 $n\in\mathbb{Z}\,,$ de fine:

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

* added by me, SammyS.

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