# Thread: Find a point set P whose first and second derived sets are nonempty and different fro

1. ## Find a point set P whose first and second derived sets are nonempty and different fro

Find a point set P whose first and second derived sets are nonempty and different from P and find each other.
I have a lot of trouble in finding this point set P.
I tried P={sin n}, and first derived set of P ={the real numbers on the interval [-1,1]}, but thus the second derived set = the first derived set.
Anybody could help me with this?
Thankss a lottt!!!

2. Originally Posted by suna
Find a point set P whose first and second derived sets are nonempty and different from P and from each other.
How about the set $\displaystyle \{(1/m,1/n)\in\mathbb{R}^2:m,n\in\mathbb{N}\}$?

3. Originally Posted by Opalg
How about the set $\displaystyle \{(1/m,1/n)\in\mathbb{R}^2:m,n\in\mathbb{N}\}$?
Thankyou, but I think the 1st derived set of this set is {0,0}, so the second derived set of it is {0}
Are they different?

4. Originally Posted by Opalg
How about the set $\displaystyle A= \{(1/m,1/n)\in\mathbb{R}^2:m,n\in\mathbb{N}\}$?
Originally Posted by suna
Thankyou, but I think the 1st derived set of this set is {0,0}, so the second derived set of it is {0}
Are they different?
Would $\displaystyle \left( {\frac{1}{2},0} \right) \in A'\,\& \,\left( {0,\frac{1}{4}} \right) \in A'$? Well of course they do!
So what is $\displaystyle {A''}$?