Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $A = \{a +\pi b : a , b \in \mathbb{Z}\}$ is a subset of $\mathbb{R}$. What will be the derived set of it?

share|improve this question
    
Derived set in what sense? –  Asaf Karagila Jul 15 '12 at 17:06
1  
I think Varun wants the set of accumulation points of $A$. –  Siminore Jul 15 '12 at 17:12
    
Yes, Sir By Derived set, I mean the set of all limit points of the set $A$. –  shane Jul 15 '12 at 17:50

1 Answer 1

up vote 2 down vote accepted

$A$ is an additive subgroup of $\Bbb R$, which is not discrete since $\pi$ is irrational. A known result about additive sub-groups of $\Bbb R$ shows that $A$ is dense in $\Bbb R$ for the usual topology. Each point which is not in $A$ is a limit point.

For the points which are in $A$, we use irrationality criteria (proposition 4 in the link).

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.