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

**suna** · October 9th 2008, 06:08 PM

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!!!

**Opalg** · October 10th 2008, 12:38 AM

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 $\{(1/m,1/n)\in\mathbb{R}^2:m,n\in\mathbb{N}\}$ ?

**suna** · October 11th 2008, 04:20 PM

Originally Posted by Opalg

How about the set $\{(1/m,1/n)\in\mathbb{R}^2:m,n\in\mathbb{N}\}$ ?

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?

**Plato** · October 11th 2008, 04:56 PM

Originally Posted by Opalg

How about the set $A= \{(1/m,1/n)\in\mathbb{R}^2:m,n\in\mathbb{N}\}$ ?

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 $\left( {\frac{1}{2},0} \right) \in A'\,\& \,\left( {0,\frac{1}{4}} \right) \in A'$ ? Well of course they do!
So what is ${A''}$ ?

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